Large elastic deformations in thin rods

Summary In this paper an appropriate analytical treatment for the determination, through exact formulae, of large elastic deformations in thin skew-curved rods is presented. This problem is associated with a system of fifteen nonlinear, ordinary, differential equations of the first order; the unknowns of the system are the final curvature and torsion functions, as well as the generalized internal forces and displacements of the rod. Subsequently, the problem of a thin cantilever circular rod subjected to terminal co-planar forces is examined and closed formulae determining its generalized displacements are obtained. Finally, the effectiveness and the potentialities of the method are demonstrated by several numerical applications.

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Übersicht In diesem Artikel wird eine analytische Methode zur Bestimmung von großen elastischen Verformungen eines schief gekrümmten Stabes durch exakte Formeln entwickelt. Dieses Problem wird durch ein System von fünfzehn nichtlinearen, gewöhnlichen Differentialgleichungen erster Ordnung beschrieben; die Unbekannten des Systems sind sowohl die endlichen Krümmungs- und Torsionsfunktionen als auch die verallgemeinerten inneren Kräfte und Verschiebungen des Stabes. Ferner wird das Problem des dünnen beidseitig gelagerten zylindrischen Stabes, welcher koplanaren Endlasten unterliegt, untersucht, und geschlossene Formeln werden erhalten. Schließlich werden die Effektivität und die Möglichkeiten der Methode durch mehrere numerische Anwendungen dargestellt.

     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

On the free shapes of elastic rods

The problem of free shape consists in finding the form that an elastic body must have in a natural state in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends, and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.     

On the theory of porous elastic rods

We consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamical nonlinear field equations of the model. Then, in the framework of linear theory, we prove the uniqueness of the solution to the associated boundary-initial-value problem. We identify the relevant field quantities from the theory of directed curves by comparison with the three-dimensional equations of straight porous rods. Finally, for orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches.     

On the stability of elastic annular rods

The stability of equilibrium of non-linearly elastic rods, whose deformations obey the classical Kirchhoff’s equations, is considered. A variational formulation of the equilibrium problem is given, and the equilibrium equations for infinitesimal deformations superimposed to a finite transformation of a rod are deduced. The stability of annular rings, in which the twisting strain is non-null, is investigated by study of the second variation of the energy functional.     

Large elastic deformations in thin cantilever rods due to concentrated loadings

In this paper the problem of large elastic deformations in cantilever thin rods subjected to concentrated loads is considered. Taking into account the incompressibility assumption of the center line and the equations relating the internal moments with the curvatures and torsion of the rod before and after the deformation, the non-linear equilibrium system, composed of six coupled differential equations of first order, is transformed to a new system of higher order. The cases of geometries of initially curved rods and their cross-sections were investigated, for which the higher order system of equations may be decoupled and solved in a closed form. Several applications of thin curved cantilever rods were made and the potentialities of the method were shown with these examples.     

Non-linear elastic analysis of thin rods subjected to bending with arbitrary kinetic conditions of their cross-sections

In this paper problems concerning the non-linear analysis of thin rods due to pure bending with constant initial curvatures and twist and with arbitrary kinetic conditions of their cross-sections are presented. Couples are not considered as being applied to the rods except at their ends. The solutions developed in this paper, which determine the curvature components and the twist of the rod after deformation, are exact in the form of elliptic integrals.     

On the elastostatics of thin periodic plates with large deflections

Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applications, these models are used to analyse a bending of periodic plates under various loadings.     

Asymptotic analysis of a thin linearly elastic plate equipped with a periodic distribution of stiffeners

We derive several models of thin plates equipped with a periodic distribution of stiffeners. Depending on the orders of magnitude of the different parameters involved, diverse situations arise, from classical Kirchhoff–Love behaviour with additional energy term to full rigidification.     

On the stability of thin periodic plates

In this paper a new modelling method of thin plates with a periodic structure along one direction and with a slowly-varying structure along the perpendicular direction in the planes parallel to the plate midplane is presented. The model is a certain generalisation of the length-scale model of periodic plates, which makes it possible to take into account the effect of periodicity cell size on the dynamics of plates with two-directional periodic structure (Jȩdrysiak and Woźniak, 1995; Jȩdrysiak, 1998). In order to describe this effect in stationary processes, e.g. a plate stability, for one-directional periodic plates the generalised model is applied in this paper. Applications to the stability problems and a comparison with effective stiffnesses models will be shown.     

Frictional elastic contact with periodic loading

Quasi-static frictional contact problems for bodies of fairly general profile that can be represented as half planes can be solved using an extension of the methods of Ciavarella and Jäger. Here we consider the tangential traction distributions developed when such systems are subjected to loading that varies periodically in time. It is shown that the system reaches a steady state after the first loading cycle. In this state, part of the contact area (the permanent stick zone) experiences no further slip, whereas other points may experience periods of stick, slip and/or separation. We demonstrate that the extent of the permanent stick zone depends only on the periodic loading cycle and is independent of the initial conditions or of any initial transient loading phase. The exact traction distribution in this zone does depend on these factors, but the resultant of these tractions at any instant in the cycle does not. The tractions and slip velocities at all points outside the permanent stick zone are also independent of initial conditions, confirming an earlier conjecture that the frictional energy dissipation per cycle in such systems depends only on the periodic loading cycle. We also show that these parameters remain unchanged if the loading cycle is changed by a time-independent tangential force, provided this is not so large as to precipitate a period of gross slip (sliding).     

Disperson of bending waves in rods with periodic parameters

An analytical-numerical method is developed for analysis of the geometric dispersion of bending waves in a rod with bending rigidity and mass per unit length varying periodically along its axis. The Bernoulli-Euler equations in the case of harmonic vibrations are reduced to a Hamiltonian system of the longitudinal coordinate. The general solution is constructed. It is expressed in terms of the matriciant of the system over one period, multipliers, and the eigenvectors of monodromy matrices. A technique is developed to determine the wave propagation constant as a function of the frequency, and the conditions of wave blanking and transmission are established. The results of solution and analysis of specific problems are presented. National University of Building and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 94–99, December, 1999.     

On the propagation of thermoelastic disturbances in thin plates and rods

When restrictions are placed on the components of stress in an elastic body the linear thermoelastic equations which govern its internal motions differ from those which hold in general. The main purpose of this paper is to draw attention to these differences and to present a simple table by means of which results referring to the propagation of general, small amplitude, thermoelastic disturbances can be converted to results for longitudinal wave propagation in a thin plate and in a thin rod. Flexural vibrations are also briefly examined. By proceeding to the low frequency limit the classical equations of motion and of heat conduction appropriate to the thin plate and the thin rod are re-derived.     

Falling of a body with a flat elastic bottom on a thin liquid layer at a small angle

Nonlinear shallow water equations and the method of matched asymptotic expansions are used to solve the problem of the impact of a box-type body with a flat bottom on a thin elastic liquid layer at a small angle in the plane formulation. It is established that, at certain values of the input parameters of the problem, the liquid pressure near the body edges becomes less than atmospheric pressure, and the liquid separates from the bottom of the box. Calculations demonstrating the influence of elastic bottom and liquid separation on the body motion are performed. It is shown that the presence of an elastic bottom significantly changes the hydrodynamic pressure distribution and can cause loads higher than in the case of a rigid body.