Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

On the Principal Boundary Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are investigated.     

General Solutions of Axial Symmetrical Ring Shells

This paper gives the general solutions of axial symmetrical ring shells for all values of slender-ness ratio.This solution is newly brought out,and can be used to solve various practical nroblems,including corrugated tubes,thermal expansion joints,Borden tubes,etc.     

The Finite-Element-Force Method for Solving Plane Problems of Elasticity

Using elements in the form of arbitrary sectors,the author has devised a plan for solving plane problems of elasticity by the force method.The method is characterized by a smaller number of nodes,a more convenient computation and a perfect adaptability to the particular shape of the region in question.     

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.     

On the Asymptotic Behaviour of the Solutions of the Equations of Linear Elastodynamics in Unbounded Domains

We prove, among other things, that if the acoustic tensor satisfies a suitable growth condition at infinity (the hyperbolicity condition) and the total initial energy is summable with a suitable weight, then the solution to the initial boundary value problem of linear elastodynamics in unbounded domains decays at infinity, at every instant, with a rate depending on the weight. Moreover, we show that the hyperbolicity condition is necessary and sufficient for the equipartition in mean of the total energy.     

The Theory of Thin-Walled Rods of Open Profile as a Result of Asymptotic Splitting in the Problem of Deformation of a Noncircular Cylindrical Shell

A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate. A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements. The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses in the cross-section.     

Unified Theory of Variation Principles in Non-linear Theory of Elasticity

The purpose of this paper is to introduce and to discuss several main variation principles in non-linear theory of elasticity——namely the classic potential energy principle,complementary energyprinciple,and other two complementary energy principles(Levinson principle and Fraeijs de Veu-beke principle)which are widely discussed in recent literatures.At the same time,the generalizedvariational principles are given also for all these principles.In this paper,systematic derivation andrigorous proof are given to these variational principles on the unified bases of principle of virtualwork,and the intrinsic relations between these principles are also indicated.It is shown that,theseprinciples have unified bases,and their differences are solely due to the adoption of different varia-bles and Legendre tarnsformation.Thus,various variational principles constitute an organized totalityin an unified frame.For those variational principles not discussed in this paper,the same frame canalso be used,a diagram is giv     

Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

Discussion on“General Solutions of Axial Symmetrical Ring Shells”

There is one point to be mentioned about the decay constant of homogeneoussolution in the paper“General Solutions of Axial Symmetrical Ring Shells”(No.3.1980 of this Journal).There is an infinite number of rootsλ.The general form ofthese roots is     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ?⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N? 2)/N,0}, in which the limits of classical solutions are also continuous in ? ^N as extended functions with values in ?₊∪{∞}. We introduce a precise class of extended continuous solutions ? _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ? _c has an initial trace in ?⁺, and (iii) that the solutions in ? _c are limits of classical solutions. Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈??, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ? ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N? 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense. As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

Investigation of the Stability Problems of Elastic Bodies Using the Method of Mathematical Theory of Elasticity

In this paper,using the equilibrium equations and boundary conditionsof elastic stability problem of Новожилов and the method of mathematicaltheory of elasticity,we solve some elastic stability problems,which werestudied byищлинскииandвоицеховская,and obtained more reason-able results than theirs.     

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.     

A self-similar solution of exponential shock waves in non-ideal magnetogasdynamics

A self similar method is used to analyze numerically the one-dimensional, unsteady flow of a strong cylindrical shock wave driven by a piston moving with time according to an exponential law in a plasma of constant density. The plasma is assumed to be a non-ideal gas with infinite electrical conductivity permeated by an axial magnetic field. Numerical solutions in the region between the shock and the piston are presented for the cases of adiabatic and isothermal flow. The general behaviour of density, velocity, and pressure profiles remains unaffected due to presence of magnetic field in non-ideal gas. However, there is a decrease in values of density, velocity and pressure in case of magnetogasdynamics as compared to non-magnetic case. It may be noted that the effect of magnetic field on the flow pattern is more significant in case of isothermal flow as compared to adiabatic flow. The effect of non-idealness, specific heat exponent and magnetic field strength on the variation of shock strength across the shock front is also investigated.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

A Compact Linear Programming Procedure for Optimal Design in Plane Stress∗

Abstract

Two-dimensional problems in plane stress are considered, with a view toward obtaining the optimal distribution of thickness under the condition that there be no collapse. Geometrical constraints on the shape of the structure are included for the purpose of meeting practical limitations.

The aim of the paper is to give a new theoretical formulation to the problem in order to effect greater savings in computer time. In particular, the number of constraints is shown to be significantly reduced, and static admissibility is guaranteed even when dealing with a reduced formulation of the problem. This is done by linearizing the yield surface and by expressing the stress vectors as linear nonnegative combinations of the vertices of the yield polyhedron, and by enforcing plastic conformity in a simple compact way.

Known static and kinematic formulations are rederived by invoking the properties of linear programming. The effectiveness of the procedure is demonstrated through applications at the end of the paper.