Second-order torsion problem of a homogeneous isotropic compressible multiply-connected elastic cylinder

For simply-connected regions, some solutions are available for the second-order torsion problem of homogeneous isotropic compressible elastic cylinders based on the theory given by Green and others. In the present paper, these theories are extended to cover the second-order torsion problem for multiply-connected regions. As an example, results for torsion of a confocal elliptical ring are given.     

Asymptotic analysis of the behavior of an elastic bar under aperiodic intensive loading

It is established that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections. A method is indicated for determining the numbers of these modes, when the effect of the longitudinal vibrations on the transverse vibrations is taken into account and when it is not taken into account. A comparison of the results obtained with results of other authors [1–7] is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 160–170, May–June, 1972.     

Asymptotic models for the discrete optimal control of the deformation of an elastic membrane

This paper considers the singularly perturbed static problem of the optimal control of the deformation of an elastic membrane by means of external loads (control without constraints) applied to several small areas distant from each other. The objective functional is equal to the sum of the square of the root-mean-square approximation error and the square of the norm of the external load. Asymptotic models are constructed using the method of matched asymptotic expansions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 131–144, September–October, 2006.     

Non-linear elastic non-uniform torsion of bars of arbitrary cross-section by BEM

In this paper an elastic non-uniform torsion analysis of simply or multiply connected cylindrical bars of arbitrary cross-section taking into account the effect of geometric non-linearity is presented employing the boundary-element(BE) method. The torque-rotation relationship is computed based on the finite-displacement (finite-rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric non-linear term often described as the “Wagner strain”. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross-section's torsional rigidity is evaluated exactly without using the so-called Saint-Venant's torsional constant. The torsional rigidity of the cross-section is evaluated directly employing the primary warping function of the cross-section depending on its shape. Three boundary-value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one, employing the Analog Equation Method (a BEM-based method), yields a system of non-linear equations from which the angle of twist is computed by an iterative process. The remaining two problems are solved employing a pure BE method. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization.     

Asymptotic models for a long,elastic cylinder

We study asymptotic expansions for the displacement field of a long elastic cylinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.Research partially supported by NSF Grant DMR-8612369.     

Equilirium configurations for an elastic bar in a hard device

Some guidelines are developed for determining the equilibrium configurations of an elastic bar loaded in a hard device.     

Buckling of a bar on an elastic base

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 106–112, September–October, 1994.     

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Some remarks on the stability of the homogeneous deformation for an elastic bar

The stability of the homogeneous deformation for an elastic bar subject to uniaxial tension is studied. It is shown that the critical extension at which the homogeneous deformation becomes unstable is a decreasing function of the aspect ratio. Furthermore, it is shown that for small aspect ratios, the homogeneous deformation need not be the global minimizer of the total energy, even though it may be a local minimizer.     

Spheroidal rigid inclusion in an elastic medium under torsion

The displacement field is determined when a rigid spheroidal inclusion is present in an infinite, isotropic and homogeneous elastic medium under torsion. The values of the force and moment are derived. The solutions for the limiting cases of a sphere and a circular disk are also presented. The analysis is based on suitable distributions of singularities on the axis of symmetry of the inclusion.     

Buckling analysis of an elastic bar using the Bubnov-Galerkin method

The factors responsible for the errors often encountered in the stability analysis of elastic systems are found by testing the Bubnov-Galerkin method for the buckling problem of a single-layer flexible elastic bar. Refined formulas are obtained for the maximum deflection of a longitudinally compressed hinged three-layer bar. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 148–156, January–February, 2008.     

Postbuckling behavior of an ideal bar on an elastic foundation

Novosibirsk. Translated from Prikladnaya Mekhanaika i Tekhnicheskaya Fizika, No. 2, pp. 130–142, March–April, 1994.     

Phase boundary motion in an elastic bar of finite length

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a finite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the moving phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail by using a finite difference scheme. The numerical results show that the phase boundary in a finite bar returns to an equilibrium state after a disturbance of finite duration, whether the two-phase material is trilinear or not.     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

Global analysis of secondary bifurcation of an elastic bar

In a three dimensional framework of finite deformation configurations, this paper investigates the secondary bifurcation of a uniform, isotropic and linearly elastic bar under compression in a large range of parameters. The governing differential equations and finite dimensional equations of this problem are discussed. It is found that, for a bar with two ends hinged, usually many secondary bifurcation points appear on the primary branches which correspond to the maximum bending stiffness. Results are shown on parameter charts. Secondary modes and branches are also calculated with numerical methods. The project supported in part by the National Natural Science Foundation of China