Asymmetric Theory Of Elasticity Research Articles

PROBLEM OF PLATE BENDING IN THE MOMENT ASYMMETRIC THEORY OF ELASTICITY

For a number of materials used in modern practice, calculations according to the classical theory of elasticity give incorrect results. To ensure the reliable operation of structures, there is a need for new theories. At present, of particular interest for practical applications is the asymmetric moment theory of elasticity. In the work, by the method of hypotheses, the three-dimensional equations of the moment asymmetric theory of elasticity are reduced to the equations of the theory of plates. The hypotheses of the theory of plates in the moment theory of elasticity are formulated on the basis of previously obtained our results of the reduction of three-dimensional equations to two-dimensional theories by a mathematical method. Just as in the classical theory of elasticity, the complete problem of the moment theory of plates is divided into two problems - a plane problem and a problem of plate bending. The equations of the plane problem have been obtained in many papers. The situation is different with the construction of the theory of plate bending in the moment theory of elasticity. In this work, for the first time, substantiated hypotheses are formulated and a consistent theory of plate bending is presented. A numerical calculation of the bending of a rectangular hinged plate is carried out according to the obtained applied theory. The calculation results are presented in the form of graphs.

Решения задачи Дирихле для уравнений асимметричной теории упругости

При решении задачи групповой классификации уравнений, описывающих движение чисто механического континуума, появились некоторые новые системы дифференциальных уравнений, которые можно использовать для описания реальных физических процессов. Одна из таких новых систем: асимметричная теория упругости. Эта система может быть использована для материалов имеющих малый модуль Юнга, а также для материалов, которые работают при нагрузках близких к критическим. В данной работе изучаются уравнения асимметричной теории упругости на основе их группового расслоения: разложения системы на автоморфную и разрешающую системы, которые являются системами дифференциальных уравнений первого порядка. Построены бесконечные серии законов сохранения для разрешающей системы уравнений и автоморфной системы. Эти законы позволили решить краевую задачу Дирихле для асимметричной теории упругости в двумерном случае. Решения построены в виде квадратур, которые вычисляются по контуру исследуемой области. When solving the problem of group classification of equations describing the motion of a purely mechanical continuum, some new systems of differential equations have appeared that can be used to describe real physical processes. One of such new systems is the asymmetric theory of elasticity. This system can be used for materials with a small Young’s modulus, as well as for materials that operate at loads close to critical. In this paper, the equations of the asymmetric theory of elasticity are studied on the basis of their group bundle: decomposition of the system into automorphic and resolving systems, which are systems of differential equations of the first order. Infinite series of conservation laws are constructed for a resolving system of equations and an automorphic system. These laws made it possible to solve the Dirichlet boundary value problem for the asymmetric theory of elasticity in the two-dimensional case. The solutions are constructed in the form of quadratures, which are calculated along the contour of the studied area.

Conformation Control of DNA Molecules by Means Geometric and Physical Parameters

A conceptual approach to the problem of managing spatial configurations of DNA molecules is considered. The work is problematic in nature and is a synthesis of the authors’ research in the field of modeling the behavior and structure of DNA by the methods of the mechanics of a deformable solid. The subject of research in this paper is the question of the applicability of methods of control theory to a living object by the example of a DNA molecule. The paper considers both issues of controllability on examples of the influence of the parameters of a molecule on its configuration, and questions of observability and identification of parameters of a molecule, based on the visible configuration in the natural environment. A brief review of the authors’ results in terms of adaptation to the objects of research of existing and development of new mathematical models of deformable elastic objects with regard to their internal structure is given. The proposed approach is based on the concept of transition using known methods of molecular dynamics from a multi-element discrete medium to a continuum containing momentary stresses. To this end, in previous works, the authors obtained the dependence of the components of the strain tensors, force and moment stresses on various types of interatomic interaction potentials (LennardJones potential, Born-Meyer potential, etc.). The need to choose as the base model of a continuum containing momentary stresses is dictated by the peculiarities of the main object of study - nucleic acid molecules and biopolymers - which have several degrees of freedom of rotational motions. Also, as an example, we consider the case for which the reduction from the three-dimensional problem of the asymmetric theory of elasticity to a one-dimensional one was carried out by splitting the three-dimensional problem into a set of two-dimensional and one-dimensional problems. The kinematic parameters that are necessary to attract in order to obtain a closed system of equations of the one-dimensional moment theory of rods with the system of Kirchhoff’s differential equations are indicated. The remaining geometrical values are found from the relations defining them. The proposed approach is consistent with current trends in the field of molecular modeling in biophysics and physico-chemical biology, and it seems promising in solving the problems of controlling genetic and biochemical processes involving DNA.

Finite element analysis of two- and three-dimensional static problems in the asymmetric theory of elasticity as a basis for the design of experiments

In this paper, the constitutive relations of the finite element method are constructed and used for solving two- and three-dimensional problems of the asymmetric theory of elasticity. Different variants of finite elements are considered. The numerical experiments are carried out to evaluate the reliability and computational efficiency of the finite element algorithm based on the comparison between the numerical and analytical solutions, numerical estimation of the convergence and checking of the degree of accuracy, to which the natural boundary conditions are satisfied. The obtained solutions to the two- and three-dimensional problems are interpreted from the viewpoint of their applicability to a design of experiments capable of revealing the facts of couple-stress effects in material under elastic deformation and identification of material constants for the asymmetric theory of elasticity. The capabilities of the finite element algorithm to interpret experimental data and estimate the errors occurring in real experiments have been tested by solving several example problems.

Determining the effective thermoelastic characteristics of regularly laminated composite in the asymmetric theory of elasticity

The asymmetric theory of elasticity is used to model a hybrid laminated composite of regular structure with all phases isotropic. The effective thermoelastic characteristics of the composite are determined. It is shown that the equations derived can be used to determine stress-strain state in all the phases of the composite using the average components of the tensors of force stresses, couple stresses, strains, and wryness in a layered material, which is of fundamental importance for the design of composites based on structural theories of failure

The problem of surface wave propagation in a reduced Cosserat medium

This article continues a series of publications devoted to the study of waves in the framework of the asymmetric theory of elasticity, where the deformed state of the medium is characterized by independent vectors of translation and rotation. The problem of acoustic Rayleigh wave propagation in half space is considered within a model of the reduced Cosserat medium. A general analytic solution of this problem is obtained. The analysis of this solution is compared with the corresponding solution for a classical elastic medium and full linear Cosserat medium. It is shown that the Rayleigh wave is characterized by a range of forbidden frequencies, where this wave cannot propagate. The dispersion curve consists of two branches. One of them has a cut-off frequency and cut-off wavenumber.

Constructing the governing equations for a layered composite of regular structure within the limits of the theory of asymmetric elasticity on the basis of energy criteria of equivalence

Within the limits of the theory of asymmetric elasticity, two (kinematic and static) approaches are offered to constructing the governing equations for a layered hybrid composite of regular structure with isotropic constituents. As a criteria of equivalence between the layered composition and a homogeneous transversely isotropic material, the equality of the specific free energy and the specific thermodynamic Gibbs energy in them is used, which allows one to determine the upper and lower bounds for the effective rigidities of the layered material.

Towards a new generation of 2D mathematical models in the mechanics of thin-walled fibre-reinforced structural components

Two-dimensional mathematical modelling of fibre-reinforced thin-walled structures is based on the simplifying assumption that fibres have negligible thickness and they are perfectly flexible. Micromechanics considerations reveal that, although this assumption is a valid approximation in many cases of interest, resistance of fibres in bending gives rise to asymmetric elasticity theory effects. A relevant finite elasticity theory developed recently [A.J.M. Spencer, K.P. Soldatos, Int. J. Non-lin. Mech. 42 (2007) 355–368] has produced, as a particular case, a version of asymmetric linearised elasticity theory that takes into consideration the effects of fibres bending stiffness. The latter, asymmetric theory of linear elasticity is currently available in a form appropriate to the material symmetries of transverse isotropy only but it can serve as a means for the incorporation of the aforementioned micromechanics effects into accurate two-dimensional thin-walled structures modelling. In this context, this study is the first attempt towards two-dimensional mathematical modelling of plate-type structural components, the fibre-reinforced material of which contains fibres that resist bending.

Boundary-value problems of the asymmetric theory of elasticity for thin plates

Boundary-value problems of the three-dimensional asymmetric micropolar, moment theory of elasticity with free rotation are considered for thin plates. It is assumed that the total stress-strain state is the sum of the internal stress-strain state and the boundary layers, which are determined in an approximation using asymptotic analysis. Three different asymptotic forms are constructed for the three-dimensional boundary-value problem posed, depending on the values of dimensionless physical constants of the plate material. The initial approximation for the first asymptotic form leads to a theory of micropolar plates with free rotation, the initial approximation for the second asymptotic form leads to a theory of micropolar plates with constrained rotation, and the initial approximation for the third asymptotic form leads to a theory of micropolar plates with “small shear stiffness.” The corresponding micropolar boundary layers are constructed and studied. The regions of applicability of each of the theories of micropolar plates constructed are indicated.

Analytical and numerical solutions for static and dynamic problems of the asymmetric theory of elasticity

The paper offers a number of new analytical solutions of static and dynamic wave problems for a linear elastic Cosserat continuum. We compare them to corresponding analytical solutions within the classical theory of elasticity and illustrate the manifestations of couple-stress effects using model materials. Concurrently with the derivation of the analytical solutions, the finite-element method is used to develop numerical solution algorithms for problems of the asymmetric theory of elasticity on static deformation. The algorithms give rise to nu-merical results that significantly expand the considered class of analytical solutions. The obtained results provide the basis for the performance of physical experiments that investigate the moment behavior of materials under elastic deformation.

Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum

The problem of propagation of an acoustic surface Rayleigh wave in an infinite half-space is considered within the framework of the asymmetric theory of elasticity (Cosserat medium). It is assumed that material deformation is described not only by the displacement vector but also by an independent rotation vector. A global analytical solution of the problem in displacements is obtained. A comparative analysis of the solution obtained and the corresponding solution for the classical elastic medium is performed. Macroparameters characterizing the difference of the stress-strain state from that predicted by the classical theory of elasticity are introduced.

Parametric analysis of analytical solutions to one‐ and two‐dimensional problems in couple‐stress theory of elasticity

AbstractIn this paper the fundamentals of the asymmetric elasticity theory are used to consider one‐ and two‐dimensional boundary‐value problems: shear deformation of elastic infinite plane layer(plate); torsion of a ring rigidly fixed at the external contour due to rotation of the inner one; deformation of a plane washer caused by a rigid displacement of the internal contour relative to the external one; the Kirsch problem on unilateral extension of a plate loosened by a circular hole. The solution of each problem is compared with the corresponding solution obtained in the framework of the classical theory of elasticity. The comparison is made in terms of macro‐parameters introduced to characterize the degree of difference between these solutions. The analysis of the obtained results show that for each problem under consideration this difference is not essential. It is worthy of note that the macro‐parameters used for comparison can be constructively measured by experiment. The obtained results can be used to outline a key diagram of experiments enabling one to detect the effects of “couple” response of the examined medium.

Variational principles of asymmetric elasticity theory of finite deformation

In this paper, based on the finite deformation S-R decomposition theorem, the definition of the body moment is renewed as the sum of its internal and external. The expression of the increment rate of the deformation energy is derived and the physical meaning is clarified. The power variational principle and the complementary power variational principle for finite deformation mechanics are supplemented and perfected.

The theory of very shallow shells, based on the asymmetric theory of elasticity

A method of reducing a three-dimensional problem in the asymmetric theory of elasticity to a two-dimensional problem in the theory of shells is proposed.

Membrane shell theory based on asymmetric elasticity theory

Membrane shell theory based on asymmetric elasticity theory

The theory of transverse bending of plates with asymmetric elasticity

The problems of bending plates made of various types of composites can be solved efficiently using Cosserat's theory. An approximate theory, reducing three-dimensional equations of the asymmetric theory of elasticity of solids to two-dimensional equations of plate theory has been formulated.

A general solution of the static problem of the theory of asymmetrical elasticity

A general solution of the homogeneous static relations of the theory of asymmetric elasticity is constructed. The passage to the solution of the classical (symmetric) theory of elasticity is shown, and the form of the general solution for the plane problem is derived. Certain modifications to the general solution of the equations of equilibrium in the theory of elasticity serve as a basis for formulating various different expressions for the Castigliano functional in the stress functions /1/.

Symmetric and Asymmetric Theories of Bimodulus Elasticity‐Bending of Cylindrical Panels

AbstractSome composite materials and high‐polymers behave or may be approximately regarded to behave as bimodulus materials. There are several investigations being conducted on the symmetry of elastic constants appearing in two‐or three‐dimensional stress‐strain relations of these materials, and some experimental results confirm their symmetry while others establish their asymmetry. In this paper, by solving a simple problem based either on the symmetric or asymmetric theory of elasticity, these theoretical differences are discussed. The resulting difference in deflection of laterally loaded cylindrical panels is less than few percent.

The boundary value problem of asymmetric elasticity theory in quasi-classical approximation: PMM vol.36, n≗2, 1972, pp.282–290

Considered is the static boundary value problem of the asymmetric theory of elasticity for media in which the moment effects [couple stresses] supply a small contribution to the elastic energy. The elastic coefficients in the equilibrium equations, having the dimensions of a squared length, are taken to be small in comparison with the squares of the characteristic dimensions of the body. One obtains the solution of the equilibrium equations, containing small parameters in the leading derivatives. By an approximation method one constructs the solution for the field of displacements and rotations in the form of the sum of their classical limits and moment terms having the form of boundary layer functions. Boundary conditions of kinematic type are considered and a scheme is developed in order to satisfy them by the method of successive approximations. Most of the media whose viscoelastic behavior is described within the limits of the asymmetric continuum mechanics (liquid) crystals, ferromagnetics, in a series of cases dislocation media and suspensions) are characterized by a small contribution of the moment terms in the general energetic balance of the deformation and flow processes. However, without taking into account the interaction of the moments one cannot give and interpretation to an entire series of delicate singularities of their viscoelastic behavior (the effects of the elastic distortion in the field of directions of the axes of molecular orientation in liquid crystals, the formation of spin waves in ferromagnetics, the effects of hardening in the plastic deformation, peculiarities of blood flow, etc.) In connection with this there aiises the problem of analyzing the simplifications which can be introduced in the asymmetric mechanics by the investigation of media with weak moments. Below we consider elastic Isotropie media which are characterized by additional coefficients of rotational elasticity λ and moment elasticity η, τ, θ [1]. The energetic contribution of the moment terms to the elastic potential is determined by the ratio between these coefficients and tine moduli λ, μ of the classical elasticity. If these ratios are small (in some sense which will be specified later), then we will say that the medium has weak moments [couple stresses]. For such media we investigate boundary value problem of the asymmetric elasticity theory.

Certain problems of the theory of asymmetric elasticity

Certain problems of the theory of asymmetric elasticity