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.     

A Material Momentum Balance Law for Rods

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin’s circulation theorem. A novel identity for the singular sources at a discontinuity is also established. Dedicated to James N. Flavin, my friend and mentor, on the occasion of his 70th Birthday.     

On Objectivity and Material Symmetry for Simple Elastic Solids

Restrictions imposed on stress response functions by objectivity and material symmetry are usually obtained by appeal to both a reference configuration and a response function common to all observers. Here these restrictions are obtained by regarding the reference configuration to be observer-dependent in a natural manner, and without the need to assume the existence of a common response function. Remarks are made on the distinction which should be drawn between the notions of observer transformation and rigid deformation, the consequences of this distinction for the correct interpretation of ‘invariance under superposed rigid motions’, and on the domains of response functions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Local Symmetry Group in the General Theory of Elastic Shells

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.     

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．     

Saint-Venant's Principle in the Asymptotic Analysis of Elastic Rods with One End Fixed

The authors take advantage of an already stated Saint-Venant's principle for a linear elastic free rod to obtain a generalization in the case of the rod fixed on one end. The result is established for star-shaped cross sections and assumes some regularity on the axial forces. Also, an exposition of the technique and its conjecture for the case of the rod fixed on both of its ends is given.     

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

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.     

Comparison between the Material Symmetry Approaches of Gurtin and Noll

The material symmetry condition of Noll is compared with that of Gurtin for elastic solids. (In this paper elastic solids refer to simply elastic solid materials.) We demonstrate that the difference between the two approaches is due to the fact that Gurtin's involves an additional rotation of the configuration. We show that the two ideas of material symmetry are equivalent for elastic solids in the context of finite elasticity theory. However, the procedures for applying these two approaches to classical linear elasticity theory are different. Also, we observe that both methods can be directly applied to the invariant infinitesimal theory of Casey and Naghdi without starting with the case of finite deformations. This revised version was published online in July 2006 with corrections to the Cover Date.     

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。     

Material Symmetry in a Theory of Continuously Defective Crystals

We propose that a particular group, which depends on the dislocation density tensor, be adopted as material symmetry group for some classes of defective crystals, and give motivation for this proposal.     

弹性涂层材料的冲蚀损伤特性研究

文章设计了材料冲蚀疲劳特性试验及通过曲线拟合确定弹性接触条件下材料冲蚀特性参数的方法，并用力学分析与冲蚀试验相结合得出弹性涂层材料的粒子冲蚀疲劳特性曲线(lgσ-lgN^*曲线)。进一步分析了弹性接触状态下冲蚀磨损率与材料特性及工况主要参数之间的定量关系。结果表明，这种材料的冲蚀疲劳极限应力σ0在MPa左右，冲蚀磨损率大约与冲击速度及冲击角的正弦成2．6次方关系。     

Paradoxical Bending Behavior of Shearable Nonlinearly Elastic Rods

In nonlinear elasticity the exact geometry of deformation is combined with general constitutive relations. This allows a very sophisticated interaction of deformations in different material directions. Based on the Cosserat theory for planar deformations of nonlinearly elastic rods we demonstrate some paradoxical bending effects caused by a nontrivial interaction of extension, flexure, and shear. The analytical results are illustrated by numerical examples.     

纤维排列方式对单向纤维加强复合材料弹性常数的影响

本文应用GMC方法计算了纤维在规则排列和随机排列时单向纤维加强复合材料的总体弹性系数，结果表明，在一定纤维体积比范围内，虽然纵向弹性模量不受纤维排列方式的影响，但是复合材料的横向弹性模量和横向剪切模量受纤维排列方式的影响较大，所以，在考察复合材料的横向弹性常数时，应该考虑纤维排列方式的影响，在相同纤维比的情况下，正方形排列和正方形对角排列所得的弹性常数是上下界，而六角形排列和随机排列的结果位于以上两者之间，当纤维比较小时（小于20％），纤维的排列方式对各弹性常数均无明显影响，本文的结果在工程应用中有指导意义。     

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.     

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.     

On Formulating and Assessing Continuum Theories of Electromagnetic Fields in Elastic Materials

My aim is to explore some ideas about the foundations of electromagnetic theory for elastic materials and to suggest some ways of assessing theories of this kind. I will describe some old ideas that seem to have been forgotten, about forces exerted by matter and fields on each other, and a similar idea about energies. Among other things, I will trace Toupin’s thinking about elastic dielectrics, showing how he moved toward using these ideas, although he did not explicitly recognize them. Further, I will explain how his dynamical theory can be interpreted to be consistent with them, although this is not obvious from what he wrote. Dedicated to the memory of Ronald Rivlin.