S. A. Lurie

Generalized theory of elasticity

We obtain elasticity equations of higher (in the general case, infinite) order than the equations of the classical theory. In contrast to the numerous known versions of the nonclassical theory (Cosserat, nonsymmetric, microstructure, micropolar, multipolar, and gradient), which also result in higher-order equations and contain elasticity relations for traditional and couple stresses with a large number of elastic constants, our theory, regardless of the order of the equations, contains only one additional constant, which can be expressed in terms of the microstructure parameter of the medium. The basic equations of the generalized theory are presented for one-, two-, and three-dimensional problems; these equations take into account the stress gradients and can be written in terms of generalized stresses, strains, and displacements. A boundary value problem that does not require the introduction of couple stresses is stated for the generalized theory of elasticity.

Eshelby integral formulas in gradient elasticity

Eshelby integral formulas play a fundamental role in mechanics of composite materials, because they provide an efficient tool for determining the average properties of dispersion-filled materials. For example, their use in the framework of the self-consistent averaging method actually gives a final and quite precise solution to the problem of determining effective physical and mechanical properties of filled composites up to large relative contents of inclusions and almost all relations between the phase characteristics of the composite. In the present paper, we generalize the Eshelby integral formulas to the gradient theory of elasticity. This provides the possibility for using efficient methods for estimating the average characteristics of micro and nano-structured materials in the framework of gradient theories, which permit taking the scale effects into account correctly, and hence find wider and wider applications in describing the mechanical and physical processes.

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered.The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.

Ideal nonsymmetric 4D-medium as a model of invertible dynamic thermoelasticity

The space-time continuum (4D-medium) is considered, and a generalized model of reversible dynamic thermoelasticity is constructed as a theory of elasticity of an ideal (defect-free) nonsymmetric 4D-medium that is transversally-isotropic with respect to the time coordinate. The definitions of stresses and strains for the space-time continuum are introduced. The constitutive equations of the medium model relating the components of nonsymmetric stress and distortion 4D-tensors are stated. Physical interpretations of all tensor components of the thermomechanical properties are given. The Lagrangian of the generalized model of coupled dynamic thermoelasticity is presented, and the Euler equations are analyzed. It is shown that the three Euler equations are generalized equations of motion of the dynamic classical thermoelasticity, and the last, fourth, equation is a generalized heat equation which allows one to predict the wave properties of heat. An energy-consistent version of thermoelasticity is constructed where the Duhamel-Neumann and Maxwell-Cattaneo laws (a nonclassical generalization of the Fourier law for the heat flow) are direct consequences of the constitutive equations.

Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.This paper is the first to present the exact analytical solution of the Eshelby–Christensen problem for the gradient theory, which permits estimating the influence of scale effects on the stress state and the effective properties of the dispersed composites under study.We also analyze the influence of scale factors.

Refined gradient theory of scale-dependent superthin rods

A version of the refined nonclassical theory of thin beams whose thickness is comparable with the scale characteristic of the material structure is constructed on the basis of the gradient theory of elasticity which, in contrast to the classical theory, contains some additional physical characteristics depending on the structure scale parameters and is therefore most appropriate for modeling the strains of scale-dependent systems. The fundamental conditions for the well-posedness of the gradient theories are obtained for the first time, and it is shown that some of the known applied gradient theories do not generally satisfy the well-posedness criterion. A version of the well-posed gradient strain theory which satisfies the symmetry condition is proposed. The well-posed gradient theory is then used to implement the method of kinematic hypotheses for constructing a refined theory of scale-dependent beams. The equilibrium equations of the refined theory of scale-dependent Timoshenko and Bernoulli beams are obtained. It is shown that the scale effects are localized near the beam ends, and therefore, taking the scale effects into account does not give any correction to the bending rigidity of long beams as noted in the previously published papers dealing with the scale-dependent beams.

New Solution of Axisymmetric Contact Problem of Elasticity

We consider two problems of elasticity, namely, the Boussinesq problem about the action of a lumped force on a half-space and the related problem about the interaction of the half-space with a cylindrical rigid punch with plane base. In the classical statement, these problems have singular solutions. In the Boussinesq problem, the displacement under the action of the force is infinitely large, and in the punch problem, the infinitely large variable is the pressure on the punch boundary. In the present paper, these problems are solved with the use of relations of generalized elasticity derived regarding a medium element of small but finite dimensions rather than a traditional infinitesimal element. The structure parameter of the medium contained in the solutions can be determined experimentally. The obtained generalized solutions of the problems under study are regular.

Modeling of the localy-functional properties of the material damaged by fields of defects

It is shown that for Mindlin media with fields of defects there is an alternative interpretation allowing to describe the material affected by defects as equivalent functionally-gradient material with varying properties for coordinates, modeled in the classical theory of elasticity. We establish clear relationships for determining the properties of functionally graded materials by the solutions, taking into account the availability of fields of defects. It is shown that, in general, the properties of equivalent functionally-gradient material depend on the coordinates, as well as on the loading and boundary conditions.

New solution of the plane problem for an equilibrium crack

We consider the classical plane problem of elasticity about a crack in an isotropic elastic unbounded plane resulting in a singular solution for the stresses near the crack edge. Relations of generalized elasticity with a small parameter characterizing the medium microstructure are derived, and the higher order of these relations permits eliminating the singularity of the classical solution. An experimental method for determining the medium parameter is proposed, and the corresponding experimental results are given.

Generalized solution of the problem on a circular membrane loaded by a lumped force

The solution of the problem on a circular membrane loaded at the center by a lumped force is a classical example of a singular solution of equations of mathematical physics. In this paper, the problem is solved by using relations of the generalized theory of elasticity, which contain a structural parameter and permit obtaining a regular solution. An experiment for determining the structural parameter in the problem of bending of a membrane is described.