Leonid P. Lebedev

Existence of weak solutions in elasticity

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.

The Rayleigh and Courant variational principles in the six-parameter shell theory

Courant’s 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. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated.

Some Issues of Continuum Mechanics and Mathematical Problems in the Theory of Thin-Walled Structures

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.

Micropolar Shells as Two-dimensional Generalized Continua Models

Using the direct approach the basic relations of the nonlinear micropo- lar 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 micropolar continuum. Each point of the micropolar shell has three translational and three rotational degrees of freedom as in the rigid body dynamics. In this theory the rotations are kinematically independent on translations. The interaction between of any two parts of the shell is described by the forces and moments only. So at the shell boundary six boundary conditions have to be given. In contrast to Kirchhoff-Love or Reissners models of shells the drilling moment acting on the shell surface can be taken into account. In the paper we derive the equilibrium equations of the shell theory using the principle of virtual work. The strain measures are introduced on the base of the principle of frame indifference. The boundary-value static and dynamic problems are formulated in Lagrangian and Eulerian coordinates. In addition, some variational principles are presented. For the general constitutive equations we formulate some constitutive restrictions, for example, the Coleman-Noll inequality, the Hadamard inequality, etc. Finally, we discuss the equilibrium of shells made of materials un-

Introduction to mathematical elasticity

Models and Ideas of Classical Mechanics Simple Elastic Models Theory of Elasticity: Statics and Dynamics.

Kinematics of Micropolar Continuum

In this chapter we briefly recall general kinematical relations for a micropolar continuum.

Mathematical Study of Boundary-Value Problems of Linear Elasticity with Surface Stresses

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 uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. The studies are performed applying the functional analysis techniques. Solutions of the problems under consideration are more smooth on the boundary surface than solutions of corresponding problems of the classical linear elasticity. The weak setup of the eigen-value 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 surface stresses are presented. These bounds demonstrate increases in both the rigidity of the body and of the eigenfrequencies over those of the body with surface stresses neglected. The considered weak statements of the initial and boundary problems constitute the mathematical foundation for some numerical methods, in particular, for the finite element method.

On Spatial Effects of Modelling in Linear Viscoelasticity

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 homogeneous. One such property is the (Lyapunov) stability of the deformation. To ensure that the material possesses these properties it is necessary to impose some additional restrictions on the model parameters. These restrictions are found by analysing the boundary value problems for viscoelastic bodies of various shapes and subjected to various boundary conditions.

Applications to Inverse Problems

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 medium.

Elements of Nonlinear Functional Analysis

From the viewpoint of functional analysis, nonlinear problems of mechanics are more complicated than linear problems; as in mechanics, they require new techniques for their study. Many of them, such as nonlinear elasticity in the general case, provide a wide field of investigation for mathematicians (see Antman [2]) ; the problem of existence of solutions in nonlinear elasticity in general is still open.