Leonid P. Lebedev

Leonid P. Lebedev
National University of Colombia | UNAL · Departamento de Matematicas

Ph.D. , D.Sc.

About

113
Publications
10,905
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1,247
Citations
Additional affiliations
May 2001 - present
National University of Colombia
Position
  • Professor (Full)
Education
September 1970 - September 1973
Southern Federal University
Field of study
  • continuum media mechanics

Publications

Publications (113)
Article
A study of existence and uniqueness of weak solutions to boundary value problems describing an elastic body with weakly nonlocal surface elasticity is presented. The chosen model incorporates the surface strain energy as a quadratic function of the surface strain tensor and the surface deformation gradients up to Nth order. The virtual work princip...
Chapter
In the framework of the strain gradient surface elasticity we discuss a consistent form of surface kinetic energy. This kinetic constitutive equation completes the statement of initial–boundary value problems. The proposed surface kinetic energy density is the most general function consistent with the constitutive relations in bulk. As the surface...
Article
Full-text available
Within the framework of the nonlinear elastic theory of micromorphic continua we derive the conditions for propagation of acceleration waves. An acceleration wave, also called a wave of weak discontinuity of order two, can be treated as a propagating nonmaterial surface across which the second derivatives of the placement vector and micro-distortio...
Article
Full-text available
Courants minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courants principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundar...
Article
Full-text available
Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin-Murdoch model to the Steigmann-Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and meth...
Article
The purpose of this paper is to use a weak setup to justify application of the finite element method (FEM) to the equilibrium problem for a nonlinear model of a shallow shell clamped along part of an edge constrained by a frictionless obstacle. A suitable energy space is constructed and the generalized (weak) solutions are introduced. The obstacle...
Article
Leonid M. Zubov was born in Yarensk, a small town near Archangelsk, in 1943. This region yielded many known Russian scientists, one of whom was Mikhail V. Lomonosov. In 1966, Zubov graduated from the faculty of physics and mechanics of Leningrad Polytechnical Institute (now Saint Petersburg State Polytechnical University), where he was later to def...
Chapter
In the theory of differential equations, inequalities are widely used to estimate or approximate solutions to problems. They are also needed to establish uniqueness and existence, along with other theoretical results pertaining to solution behavior. The purpose of this chapter is to touch on a few inequalities that play key roles in the study of di...
Chapter
Inequalities lie at the heart of mathematical analysis. They appear in the definitions of continuity and limit (and hence in the definitions of the integral and the derivative). They play crucial roles in generalizing the notions of distance and vector magnitude. But many problems of physical interest also rely on simple inequality concepts for the...
Chapter
Here we examine certain famous inequalities that have left bold imprints on both pure and applied mathematics. These results, some of which are very old, pertain to functions, sequences, and integrals. We recall that integral inequalities are frequently deduced by establishing the corresponding result for series, writing it out for Riemann sums, an...
Chapter
Some major advances in mathematics have occurred through the extension of existing number systems. The natural numbers were extended to the real numbers, the real numbers to the complex numbers, and so on.
Chapter
Generality is gained by working in abstract spaces. For instance, all essential aspects of the topics of convergence and continuity can be studied in the context of a metric space. When we search for solutions to problems of physical interest, we must often search among the members of linear spaces (also known as vector spaces). Inequalities provid...
Chapter
In this chapter we revisit some facts from mathematical analysis and show how these may be used to establish important inequalities. We begin by reviewing convergence of real number sequences and continuity of real functions of a single variable.
Chapter
The reader who has worked patiently through the mathematical content of the previous chapters should be comfortable dealing with the applications treated here. These topics were chosen for variety and are presented in no particular order (just as we might encounter them in practice).
Article
Full-text available
This paper concerns with existence and uniqueness of a weak solution for elliptic systems of partial differential equations with mixed boundary conditions. The proof is based on establishing the coerciveness of bilinear forms, related with the system of equations, which depend on first-order derivatives of vector functions in Rn. The condition of c...
Chapter
From a functional analytic standpoint, nonlinear problems of mechanics are more complicated than linear problems; as in mechanics, they require new approaches. Many, like the problems of nonlinear elasticity in the general case, provide a wide field of investigation for mathematicians (see Antman [2]); the problem of existence of solutions in nonli...
Chapter
In this chapter we briefly recall general kinematical relations for a micropolar continuum.
Chapter
Full-text available
Following [1, 2] a mathematical investigation of initial-boundary and boundary-value problems of statics, dynamics and natural oscillations for elastic bodies including surface stresses is presented. The weak setup of the problems based on mechanical variational principles is given with introducing of corresponding energy spaces. Theorems of unique...
Article
Full-text available
Solvability and uniqueness of solutions to the problems of equilibrium, vibration and dynamics in a weak setup for classical and nonclassical models of linear elasticity are established in a unified framework sufficiently flexible to accommodate new elastic models.
Book
Introduction.- Metric, Banach, and Hilbert Spaces.- Mechanics Problems from the Functional Analysis Viewpoint.- Some Spectral Problems of Mechanics.- Elements of Nonlinear Functional Analysis.- Summary of Inequalities and Imbeddings.- Hints for Selected Problems.- References.- In Memoriam: Iosif I. Vorovich.- Index.-
Chapter
In this chapter using the balance of momentum and balance of moment of momentum (Euler’s laws of motion) we introduce the stress and couple stress tensors. Then we derive the motion equations of the micropolar continuum which contains the motion equations of simple (non-polar) continuum as a special case.
Chapter
In this chapter we will consider acceleration waves in nonlinear thermoelastic micropolar continua. We will establish kinematic and dynamic compatibility relations for a singular surface of second order in the media. We also will derive an analogue to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor. The condition for ac...
Chapter
Full-text available
For an arbitrary part of the body, Eqs. (3.30) and (3.31) express the balance equations for the moment and the moment of momentum. These six scalar equations contain 18 unknown quantities that are the components of tensors \(\mathbf{ T} \) and \(\mathbf{ M} \). The dependence of \(\mathbf{ T} \) and \(\mathbf{ M} \) on medium deformations is determ...
Chapter
Consider a set of particles P 1, …, P n. To locate these particles in the space ℝ3, we need a reference system. Let the Cartesian coordinates of particle P i be (ξi, ηi , ζi ). Identifying (ξ1, η1, ξ1) with the triple (x1, x2, x 3), (ξ2, η2, ζ2) with (x,4, x,5, x,6), and so on, we obtain a vector x of the Euclidean space ℝ3n with coordinates (x1, x...
Chapter
In the past, an engineer could calculate mechanical stresses and strains using a pencil and a logarithmic slide rule. Modern mechanical models, on the other hand, are nonlinear, and even the linear models are complicated. Numerical methods in structural dynamics cannot be applied without computers running specialized programs. However, a researcher...
Chapter
We obtain a spectral problem by formally considering a solution u of the form
Book
Advanced Engineering Analysis is a textbook on modern engineering analysis, covering the calculus of variations, functional analysis, and control theory, as well as applications of these disciplines to mechanics. The book offers a brief and concise, yet complete explanation of essential theory and applications. It contains exercises with hints and...
Article
A mathematical investigation of the eigenvalue problems for elastic bodies including surface stresses is presented. Weak setup of the problems is based on the Rayleigh variational principle. Certain spectral properties are established for the problems under consideration. In particular, bounds for the eigenfrequencies of an elastic body with surfac...
Chapter
Numerous specialized books and papers have been written about the subject of stability in mechanics. Most of these concentrate on methods for obtaining critical values of certain parameters and typically contain algorithms and graphs generated for describing important but very specific problems. In the present paper we take a step back and discuss...
Article
Theorems regarding existence and uniqueness of weak solutions to mixed boundary value problems in the linear theory of micropolar shells in statics and dynamics are proved. Convergence of FEM for the static mixed problems is established. Eigenvalue problems for micropolar shells are studied and properties of the spectrum and eigenmodes are formulat...
Chapter
Full-text available
Using the direct approach the basic relations of the nonlinear micropolar shell theory are considered. Within the framework of this theory the shell can be considered as a deformable surface with attached three unit orthogonal vectors, so-called directors. In other words the micropolar shell is a two-dimensional (2D) Cosserat continuum or micropola...
Book
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies....
Article
Full-text available
Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave’s propagation is for...
Article
The mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented. Weak setup of the problems based on mechanical variational principles is studied. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formula...
Article
This book provides the general reader with an introduction to mathematical elasticity, by means of general concepts in classic mechanics, and models for elastic springs, strings, rods, beams and membranes. Functional analysis is also used to explore more general boundary value problems for three-dimensional elastic bodies, where the reader is provi...
Article
Full-text available
Resumen. Se consideran algunos problemas de deformación de cascarones elásticos de la teoría lineal con el fin de determinar cuáles son las herra-mientas de matemáticas que se utilizan en mecánica y cómo influye la mecánica en las matemáticas. Palabras claves. Cascarones elásticos, mecánica, problemas con condi-ciones de frontera, existencia y unic...
Article
Full-text available
The conditions for propagation of accelerating waves in a general nonlinear thermoelastic micropolar media are established. Deformation of micropolar media is described by the time-varying displacement vector r(t) and tensor of microrotation r(t) at each point. We call a surface S(t) an accelerating wave (or a singular surface for a solution of the...
Article
Full-text available
The stability of von Mises truss is investigated when the stress¿strain diagram of the material constituting the rods displays hysteresis. This behavior, known as pseudo-elasticity, is common in such materials as filled rubbers and shape memory alloys. Loading diagrams are presented; these show that the upper and lower critical loads depend on the...
Article
Existence of generalized solutions of contact problems for a nonlinear shallow shell with a rigid obstacle is demonstrated. A mathematical model accounting for the presence of obstacle is proposed. The solution is a minimizer of the total energy functional on the set of admissible displacements. The problem reduces to the solution of a variational...
Article
En este trabajo se discuten algunas cuestiones de la teoría de problemas de contorno para la ecuación de Poisson mediante el uso de una membrana como un objeto descrito por la mencionada ecuación. Se muestra cómo una interpretación mecánica de la ecuación de Poisson permite explicar ciertas relaciones conocidas de la teoría general y también cómo p...
Chapter
This chapter aims to present in more detail some results of the theory of linear operators. We cannot pretend to give a full treatment of this vast area, and shall select only those parts which are useful in the applications under consideration. Of course, we are forced to give some general theoretical background.
Chapter
Consider a set of particles P i , i = 1,..., n. To locate these particles in the space E3, we need a reference system. Let the Cartesian coordinates of Pi be (ξ i , η i , ζ i ) for each i. Identifying (ξ i , η i , ζ i ) with (x 1, x 2, x 3), (ξ i , η i , ζ i ) with (x 4, x 5, x 6), and so on, we obtain a vector x of the Euclidean space ℝ3n with coo...
Article
In recent decades, engineers and physicists have shown an increasing interest in functional analysis and its applications. As many of these practitioners lack special training in mathematics, they sometimes run into trouble when trying to use the tools of this powerful branch of knowledge. Our purpose is to outline the connection between the tradit...
Article
Consideration is given to some issues of continuum mechanics and mechanical problems arising in the theory of thin plates and shells. The main research areas are analyzed. The results obtained in the linear and nonlinear theories of plates and shells are reviewed and some open issues and unsolved problems of those theories are formulated.
Book
This book started its life as a series of lectures given by the second author from the 1970’s onwards to students in their third and fourth years in the Department of Mechanics and Mathematics at Rostov State University. For these lectures there was also an audience of engineers and applied mechanicists who wished to understand the functional analy...
Article
The general initial-boundary value problem of the nonlinear theory of viscoelastic shells of Koiter type is considered in curvilinear coordinates. A generalized solution to the problem is defined on the basis of mechanical variational principles. By the application of Faedo-Galerkin method, we establish two theorems on the existence of generalized...
Article
The problem of the equilibrium of a non-linear plate reinforced with stiffeners is considered. The idea of a generalized solution of the problem as a critical point of the energy functional of an elastic system is introduced and the existence of a generalized solution of the problem is proved. The convergence of Ritz' method within the framework of...
Article
A study was conducted to determine the time process due to thermoelastic temperature change, called the AI passage, as a factor affecting the precision of measurements. To illuminate the problem, the influence of the AI passage on the precision of rod force transducer was considered. This transducer was demonstrated to be practically indifferent to...
Article
The problem of equilibrium of a plate with stiffening ribs is considered. A notion of a generalized solution of the problem as the critical point of the functional of the elastic system energy is introduced. Existence of the generalized solution is proved. The Ritz method convergence within the framework of this problem is substantiated.
Article
The continuity of the dependence of the non-singular solution on small perturbations of the dimensions and form of the shell is proved using methods described earlier [1]. These perturbations lead to a change in the region into which the middle surface of the shell is mapped (for example, an increase or decrease in the aperture angle of a shallow s...
Article
Full-text available
A new class of boundary value problems is presented. These problems are described by related equations of different nature and possess such properties as the appearance of highest derivatives in boundary conditions. Such problems appear to model common engineering constructions composed of elements of different mechanical natures like plates, shell...
Article
The parameters of a linear model of a viscoelastic material are determined by testing the material in homogeneous (i.e. spatially constant) states. Some of the qualitative properties of the behaviour of the material observed in the tests may be unexpectedly lost if the material is confined, so that the behaviour varies in space and is thus not homo...
Chapter
We introduced the term compact for a set S ⊂ ℝ in Definition 1.1.9; we generalized it for a set S ⊂ ℝN , and proved the Bolzano-Weierstrass theorem (Theorems 1.1.1, 1.1.2) which states that a set S ⊂ ℝN is compact iff it is closed and bounded
Chapter
Consider a perfectly elastic rod of length l, cross-sectional area A(x), Young’s modulus E (named after Thomas Young (1773–1829), undergoing longitudinal displacement u(x). There is only a single strain ∈xx = u (x) and a single stress σxx = E∈xx = Eu (x) so that its strain energy $$ U = \frac{1}{2}\int_{0}^{1} {EA(x){{{[u'(x)]}}^{2}}dx.}
Chapter
Most problems in mechanics and physics have the form ‘Find the effect of this cause.’ There are numerous examples: Find how this structure is deformed when these forces are applied to it. Find how heat diffuses through a body when a heat source is applied to a boundary. Find how waves are bent, or absorbed, as they pass through a nonhomogeneous med...
Chapter
This chapter aims to present some results from the theory of linear operators. We cannot pretend to give a full treatment of this vast field; we shall select only those parts which we shall use in later applications.
Chapter
In continuum mechanics we often encounter operator equations of the form $$x - A(\mu )x = f,$$ (7.1.1) in a Banach space X, where A(µ) is a linear operator depending on a real or complex parameter µ. The most important example is the equation governing the steady vibration of an elastic body with frequency w = λ1/2, namely $$\lambda x - Ax = f.$$ (...
Chapter
If we want to know whether a room holds enough chairs to seat some people standing outside we can do one of two things: ● Count the number of chairs, n, and the number of people, p, and see whether n ≥ p. ● Start seating the people,and continue until all the chairs are filled, or all the people are seated, whichever comes first.
Chapter
A book must start somewhere. This is a book about a branch of applied mathematics, and it, like others, must start from somebody of assumed knowledge, otherwise, like Russell and Whitehead’s Principia Mathematica it will have to start with the definitions of the numbers 1, 2 and 3. This first chapter is intended to provide an informal review of som...
Article
The proof is obtained for a theorem of the existence of the solution of a shallow shell equilibrium problem under the most general edge fixing conditions. For instance, it is sufficient that the fixing conditions should ensure that there are no displacements of the shell as a rigid whole. However, certain constraints on the magnitude of the tangent...
Article
A “nonenergetic” formulation of the boundary value problems of statics of an elastic strip based on the principle of admissible displacements, is studied. The formulation makes possible, in particular, the study of problems concerning the strips of infinite energy, while retaining the external form of the “energetic” formulation /1–3/, and produces...
Article
Theorems concerning the solvability of two problems of nonlinear theory of anisotropic plates are proved. Problem 1 concerns the equilibrium of a plate clamped at three points and acted upon by an external load, and Problem 2 concerns the equilibrium of a plate free from geometrical constrains, under the action of external forces. Solvability of ce...

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