Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods

Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ω^ε having cross section with diameter of order ε, and let u ⁽⁰⁾ –Bernoulli–Navier displacement – and u ⁽²⁾ be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u ⁽⁰⁾ + ε² u ⁽²⁾) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the rod. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Material Symmetry of Elastic Rods

This paper presents a treatment of material symmetry for hyperelastic rods. The rod theory of interest is based on a Cosserat (or directed) curve with two director fields, and was developed in a series of works by Green, Naghdi and several of their co-workers. The treatment is based on Murdoch and Cohen's work on material symmetry of Cosserat surfaces. Two material symmetry groups are discussed: one pertains to the strain-energy function, while the other pertains to the response functions. The paper closes by showing how the treatment relates to the form-invariant approach used in Green and Naghdi's papers and a treatment proposed recently by Cohen. This revised version was published online in July 2006 with corrections to the Cover Date.     

Elastic Rods with Moderate Roration

Based on the theory of a directed or Cosserat curve of Green, Naghdi and several of their co-workers, several properly invariant approximate theories of an elastic rod are developed. The approximate theories are applicable when a rotation tensor associated with the motion of the rod is moderate. In one of these theories, some of the strains are moderate while others are small. A second theory where all of the strains are small is also developed. This revised version was published online in August 2006 with corrections to the Cover Date.     

有限变形下线弹性裂尖场的渐近分析

对有限变形下线弹性Ⅰ型裂纹场建立了无需分区的统一控制方程并进行了渐近分析, 利用“打靶法”得到位移场在物质描述与空间描述下的渐近阶次分别为3／4和1,Green应变、第二类P－K应力及Cauchy应力在物质描述与空间描述下的渐近阶次分数为－1／2和－2／3;对不同泊松比,裂尖有限变形线弹性场的位移均以UⅡ或u2为主导,裂纹张开角为π,现时构形中的大变形区为一垂直初始构形中裂纹表面的狭长带状区,应力则处于由σ22主导的单向拉伸状态,角分布函数U^-Ⅱ（0）及σ22^-(0)具有奇异性,但UL^-‘(Θ）／UⅡ^－‘(0)及σij^-(θ）/σ22^-(0)均趋于有限值。     

平面杆系结构弹性波传播的实验研究

利用ＳＨＰＢ装置，用空气枪加载就３５ＣｒＭｎＳｉ钢组成的平面杆系结构受冲击载荷作用的弹性波传播进行了实验研究，给出了一些力学现象，并利用广义特征线法给出了理论与实验的比较曲线，得到了一些有益的结论．     

Stability of n-covered Circles for Elastic Rods with Constant Planar Intrinsic Curvature

A stability index is computed for the n-covered circular equilibria of inextensible-unshearable elastic rods with constant planar intrinsic curvature û and constant values for the twisting stiffness and two bending stiffnesses. A simple expression is derived for the index as a function of û, (the ratio of bending stiffness out of the plane of curvature to bending stiffness in the plane of curvature), and (the ratio of twisting stiffness to bending stiffness in the plane of curvature). In particular, for intrinsically straight rods (û = 0) we prove that the 1-covered circle is stable if and only if 1, and the n-covered circle (n>1) is stable if and only if >1, >1, and

杆系结构受横向冲击载荷下弹性波传播的实验研究

利用 Ｓ Ｈ Ｐ Ｂ装置,用空气枪加载就 ３５ Ｃｒ Ｍｎ Ｓｉ钢组成的 Ｌ型杆系结构受横向冲击载荷作用的弹性波传播进行了实验研究,给出了一些力学现象,并与轴向冲击的实验进行了对比,得到了有益的结论。     

Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells II. Incompressible Shells

This paper treats the eversion of axisymmetric, strictly convex, incompressible nonlinearly elastic shells within a general geometrically exact theory in which the shell can suffer flexure, shear, and both midsurface and transverse extension. The governing equations differ considerably from those for compressible shells. We first formulate the governing equations carefully, showing how to handle the 3-dimensional notion of incompressibility, and paying special attention to the constitutive equations. We prove that when a thickness parameter is sufficiently small, there is an everted state, having a lip near the edge, that can be approximated effectively by an asymptotic series whose error we estimate.     

The Dynamics of Stretchable Rods in the Inertial Case

Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods under tension with inertial and dissipative dynamics are derived. In the inertial case localized solutions to the amplitude equations are derived and a linear stability criterion for the pulse solutions is obtained using the Hamiltonian formulation of the problem.     

The effect of a variation in the elastic moduli on Saint-Venant's principle for a half-cylinder

A semi-infinite prismatic cylinder composed of a linear anisotropic classical elastic material is in equilibrium under zero body force and either zero displacement or zero traction on the lateral boundary. The elastic moduli become perturbed. Under suitable conditions on the base load decay estimates are derived for the difference between corresponding quantities in the unperturbed and perturbed bodies. The amplitude in each estimate involves a multiplicative factor that tends to zero as the perturbation tends to zero. The analysis, based upon a first-order differential inequality, introduces apparently new modifications of Korn's inequalities of the first and second kind.     

Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids

Methods are presented for obtaining exact analytical representations of supercoiled equilibrium configurations of impenetrable elastic rods of circular cross-section that have been pretwisted and closed to form rings, and a discussion is given of applications in the theory of the elastic rod model for DNA. When, as here, self-contact is taken into account, and the rod is assumed to be inextensible, intrinsically straight, transversely isotropic, and homogeneous, the important parameters in the theory are the excess link Δℒ (a measure of the amount the rod was twisted before its ends were joined), the ratio ω of the coefficients of torsional and flexural rigidity, and the ratio d of cross-sectional diameter to the length of the axial curve C. Solutions of the equations of equilibrium are given for cases in which self-contact occurs at isolated points and along intervals. Bifurcation diagrams are presented as graphs of Δℒ versus the writhe of C and are employed for analysis of the stability of equilibrium configurations. It is shown that, in addition to primary, secondary, and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the axial curve is a circle, there are families of equilibrium configurations that are isolas in the sense that they are not connected to bifurcation branches by paths of equilibrium configurations compatible with the assumed impenetrability of the rod. Each of the isolas found to date is connected to a bifurcation branch by a path which, although made up of solutions of the governing equations, contains regions on which the condition of impenetrability does not hold. This revised version was published online in July 2006 with corrections to the Cover Date.     

Twisted Elastic Rings and the Rediscoveries of Michell's Instability

Elastic rings become unstable when sufficiently twisted. This fundamental instability plays an important role in the modeling of DNA mechanics and in cable engineering. In 1962, Zajac computed the value of the critical twist for the instability. This critical value was rediscovered in 1979 by Benham and independently by Le Bret in elastic models for DNA; unstable rings have since become an important example of elastic instabilities in rods both for the development of new methods and in applications. The purpose of this note is to show that the problem had been completely solved by John Henry Michell in 1889 in a rather elegant manner and to reflect on its history and modern developments.     

正交异性板条的渐近分析

本文阐明由三维薄弹性体的渐近分析导出各级精度板壳理论的基本方法。将多尺度分析用于板的内部区域和边界层区域导出应力,应变和位移等物理量的不同的无量纲小厚度参数ε的渐近展开式。与工程的方法不同,推导仅基于ε→0的渐近分析,对板的变形不做任何假定。给出正交异性板的平面应变渐近展开的具体列式,仅有一些常数待定。结果表明,内域解渐近级数的首项正是熟知的Kirchhoff板理论解。本文为内域解和边界层解的渐近匹配,从而正确表述圣维南原理并用以建立各级渐近解的边界条件进而求解的研究做好了准备。     

The Solution of Saint-Venant's Problem in the Theory of Cosserat Shells

This paper is concerned with the linear theory of thin elastic shells modelled as Cosserat surfaces. We formulate the corresponding Saint-Venant's problem with respect to cylindrical shells. Then, we determine the solution of the relaxed problem for both open and closed cylindrical surfaces. The edge curves perpendicular to the generator are not necessarily circular. Finally, some particular cases are discussed.     

Analysis of Plane Phase Strains of Rods and Plates

The problem of a rod and a plate subjected to plane strain in the interval of the direct phase transformation is formulated as a nonlinear boundary-value thermoelastic problem with an implicit dependence on temperature (through a phase parameter simulating the volume fraction of new-phase crystals). An analytical solution of the problem of a rod bent into a ring and a plate bent into a tube as a result of phase strains under the action of a small end bending moment is given. A numerical analysis of the buckling problem of a titanium nickelide (alloy) rod (plate) under longitudinal compression in the interval of the direct phase transformation shows that buckling becomes possible if the compressive load is much lower than the Euler critical load calculated before the transformation. Branches of buckled equilibrium states corresponding to loads lower than the Euler load are plotted as functions of the phase parameter. In all cases considered, the deflections increase abruptly in the neighborhood of the critical points. The evolution of buckling modes is studied, and the phase-strain distributions along the rod (plate) are shown. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 156–164, March–April, 2006.     

Minimum Energy Characterizations for the Solution of Saint-Venant's Problem in the Theory of Shells

The linear theory of Cosserat surfaces is employed to study Saint-Venant's problem for cylindrical shells of arbitrary cross-section. We prove minimum energy characterizations for the solution of the relaxed Saint-Venant's problem previously obtained. Then, we determine the global measures of strain appropriate to extension, bending, torsion and flexure for certain classes of solutions to the relaxed problem. Mathematics Subject Classifications (2000) 74K25, 74G05.     

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.     

Dispersion of Small Amplitude Waves in a Pre-Stressed, Compressible Elastic Plate

The dispersion of harmonic waves, propagating along a principal direction in a pre-stressed, compressible elastic plate, is investigated in respect of the most general isotropic strain-energy function. Different cases, dependent on the choice of material parameters and pre-stress, are analysed. A complete long and short wave asymptotic analysis is carried out, with the approximations obtained giving phase speed (and frequency) as explicit functions of wave and mode number. Various wave fronts, both associated with the short wave limit of harmonics and arising through the combination of harmonics in a narrow wave speed region, are discussed. It is mentioned that the case of high compressibility is of particular interest. In contrast with the classical (un-strained) case, the longitudinal body wave speed may be less than the corresponding shear wave speed. In consequence, the short wave limit of all harmonics may be the appropriate longitudinal wave speed; contrasting with the classical case for which this limit is necessarily associated with a shear wave front. A further possible short wave limit is also shown to exist for which the associated wave normal has a component in the direction normal to the plate. Particularly novel numerical results are presented when the longitudinal and shear wave speeds are equal. The analysis is illustrated by numerical calculations for various strain-energy functions.     

ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS

Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.     

A Refined Asymptotic Model of Fluid-Structure Interaction in Scattering by Elastic Shells

A refined asymptotic model of fluid-structure interaction in scattering by elastic shells is proposed. The model takes into consideration transverse compression of a shell by a fluid and some other phenomena. As an illustration, scattering of a plane acoustic wave by a circular cylindrical shell is considered. Comparison of numerical data corresponding various approximate approaches is provided. This revised version was published online in July 2006 with corrections to the Cover Date.