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.     

On Transverse and Rotational Symmetries in Elastic Rods

The influences of transverse and rotational symmetries on the strain-energy functions of elastic rods are discussed. Complete function bases are presented and, for some constrained theories, these bases are also proven to be irreducible. The treatment of symmetry is based on a reformulation of a recent work by Luo and O’Reilly. It is also shown how this work relates to existing treatments by Antman and Healey for a particular constrained rod theory.     

Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Strong Ellipticity of the Anisotropic Linearly Elastic Materials

In this paper we derive necessary and sufficient conditions for strong ellipticity in several classes of anisotropic linearly elastic materials. Our results cover all classes in the rhombic system (nine elasticities), four classes of the tetragonal system (six elasticities) and all classes in the cubic system (three elasticities). As a special case we recover necessary and sufficient conditions for strong ellipticity in transversely isotropic materials. The central result shows that for the rhombic system strong ellipticity restricts some appropriate combinations of elasticities to take values inside a domain whose boundary is the third order algebraic surface defined by x ²+y ²+z ²−2xyz−1=0 situated in the cube , , . For more symmetric situations, the general analysis restricts combinations of elasticities to range inside either a plane domain (for four classes in the tetragonal system) or in an one-dimensional interval (for the hexagonal systems, transverse isotropy and cubic system). The proof involves only the basic statement of the strong ellipticity condition.      

On Extension and Torsion of Strain-Stiffening Rubber-Like Elastic Circular Cylinders

This paper is concerned with investigation of the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic nonlinearly elastic materials. Our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two particular phenomenological constitutive models for such materials that reflect limiting chain extensibility at the molecular level. The axial stretch γ and twist that can be sustained in cylinders composed of such materials are shown to be constrained in a coupled fashion. It is shown that, in the absence of an additional axial force, a transition value γ=γ _t of the axial stretch exists such that for γ<γ _t , the stretched cylinder tends to elongate on twisting whereas for γ>γ _t , the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. We also obtain results for materials modeled by the well-known exponential strain-energy widely used in biomechanics applications. This model reflects a strain-stiffening that is less abrupt than that for the limiting chain extensibility models. Surprisingly, it turns out that the results in this case are somewhat more complicated. For a fixed stiffening parameter, provided that the stretch is sufficiently small, the stretched bar always tends to elongate on twisting in the absence of an additional axial force. However, for sufficiently large stretch, the cylinder tends to shorten on undergoing sufficiently small twist but then tends to elongate on further twisting. These results are of interest in view of the widespread use of exponential models in the context of the mechanics of soft biological tissues. The special case of pure torsion is also briefly considered. In this case, the resultant axial force required to maintain pure torsion is compressive for all the models discussed here. In the absence of such a force, the bar would elongate on twisting reflecting the celebrated Poynting effect.      

Partial Constraint Singularities in Elastic Rods

We present a unified classical treatment of partially constrained elastic rods. Partial constraints often entail singularities in both shapes and reactions. Our approach encompasses both sleeve and adhesion problems, and provides simple and unambiguous derivations of counterintuitive results in the literature. Relationships between reaction forces and moments, geometry, and adhesion energies follow from the balance of energy during quasistatic motion. We also relate our approach to the balance of material momentum and the concept of a driving traction. The theory is generalizable and can be applied to a wide array of contact, adhesion, gripping, and locomotion problems.     

弹性椭圆夹杂纵向剪切问题

获得纵向剪切下弹性椭圆夹杂问题的精确解。将复变函数的分区全纯函数理论，Cauchy型积分和Riemann边值问题相结合，求得各复势函数之间的解析关系，从而得到问题的封闭形式解，并给出了界面应力的解析表达式。本文解答与已有文献结果一致。本文发展的分析方法，为求解复杂多连通域的平面弹性问题提供了一条有效途径。     

Nonlinear Bending of thin Elastic Rods

Exact solutions of the problem of nonlinear bending of thin rods under various fixing conditions and point dead loads are obtained. The solutions written in a unified parametric form and expressed in terms of the elliptic Jacobi functions are classified. These solutions depend on a single parameter — modulus of elliptic functions.     

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Postbuckling Behavior of Compressed Rods in an Elastic Medium

The postbuckling of rods loaded by a compressive force P in an elastic medium is considered. The resolving nonlinear equation is obtained, and a method for solving this equation is given. It is shown that, for large lengths, in contrast to the case without elastic medium, the deflection increases as the force P decreases after the loss of stability. Several simple finite-element models, namely, the problems of compression of multilink rods with links connected by springs, are considered to confirm this effect.     

Physically Nonlinear Ellipsoidal Inclusion in a Linearly Elastic Medium

This paper considers a physically nonlinear ellipsoidal inclusion in an elastic space loaded at infinity by uniform external forces. Relations are obtained that link the stresses and strains at infinite points of the medium and in the inclusion (in the latter, a homogeneous stress–strain state occurs). Some examples, in particular, inclusions in the shape of oblate and prolate spheroids exhibiting nonlinear creep properties, are discussed.     

Regularity for Shearable Nonlinearly Elastic Rods in Obstacle Problems

Based on the Cosserat theory describing planar deformations of shearable nonlinearly elastic rods we study the regularity of equilibrium states for problems where the deformations are restricted by rigid obstacles. We start with the discussion of general conditions modeling frictionless contact. In particular we motivate a contact condition that, roughly speaking, requires the contact forces to be directed normally, in a generalized sense, both to the obstacle and to the deformed shape of the rod. We show that there is a jump in the strains in the case of a concentrated contact force, i.e., the deformed shape of the rod has a corner. Then we assume some smoothness for the boundary of the obstacle and derive corresponding regularity for the contact forces. Finally we compare the results with the case of unshearable rods and obtain interesting qualitative differences. (Accepted January 21, 1998)     

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.     

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．     

Wavelet Analysis of a Single Pulse in a Linearly Elastic Composite

Wavelet analysis is applied to study the evolution of a solitary wave propagating in a microstructural material (composite). Representing the solution in terms of elastic wavelets makes it possible to reveal the main wave microstructural effects and to extend the class of initial pulses. The initial profile is taken from short-term load experiments__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 38–46, April 2005.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

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.