Euler-Lagrange Equations for Nonlinearly Elastic Rods with Self-Contact

 We derive the Euler-Lagrange equations for nonlinearly elastic rods with self-contact. The excluded-volume constraint is formulated in terms of an upper bound on the global curvature of the centre line. This condition is shown to guarantee the global injectivity of the deformation of the elastic rod. Topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed, e.g., in DNA-molecules, are included by using the notion of isotopy and Gaussian linking number. The bound on the global curvature as a nonsmooth side condition requires the use of Clarke's generalized gradients to obtain the explicit structure of the contact forces, which appear naturally as Lagrange multipliers in the Euler-Lagrange equations. Transversality conditions are discussed and higher regularity for the strains, moments, the centre line and the directors is shown. (Accepted December 20, 2002) Published online April 8, 2003 Communicated by S. S. Antman     

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.     

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.     

Contact with a Corner for Nonlinearly Elastic Rods

In elastic contact problems it is usually required that the contact force has to be directed normally to the contact surface in the absence of friction. For an obstacle with nonsmooth surface this gives infinitely many normal directions at an edge or at a corner. For the case where a nonlinearly elastic rod under terminal loads is hanging over a needle, it is shown that the balance equations supplemented with such a normality condition have a continuum of solutions. Moreover, an additional contact condition is derived from a corresponding variational problem by means of special inner variations that preserve the shape of the rod. This way one is finally lead to a unique solution at least locally.     

Stress Solution for Transversely Isotropic Corrugated Hollow Cylinders

The stress problem for corrugated hollow transversely isotropic cylinders is solved in three-dimensional formulation for certain end conditions. Discrete Fourier series are used to make the problem one-dimensional, which is then solved by the stable discrete-orthogonalization method. Examples of analysis are given__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 62–69, March 2005.     

边坡预应力锚杆蠕变的数值分析

运用非线性蠕变分析理论模型,采用MSC.MARC软件重点研究了锚杆的应力指数、蠕变系数和几何参数的变化对于锚杆应力松弛的影响.通过对预应力锚杆支护的边坡进行数值分析,探讨了锚杆的蠕变规律以及锚杆预应力松弛现象.锚杆的参数取不同数值时,比较数值模拟结果发现:锚杆的轴力和剪力主要分布在锚固段的前端;随着锚杆应力指数及蠕变系数的增加,蠕变应变增大,应力松弛加快.     

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)     

Dynamics of Linear Piezoelectric Rods

A one-dimensional model of a linear piezoelectric thin rod is deduced from three-dimensional piezoelectricity by introducing suitable internal constraints and appropriate hypotheses on the electric displacement field.     

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.     

Existence of Exact Solutions for the Eversion of Elastic Cylinders

The existence of eversion deformations of an elastic cylinder into another right circular cylinder is studied. It is found that such deformations are associated with strain-energy functions of separable form W(λ₁,λ₂,λ₃) =φ(λ₁)+φ(λ₂)+φ (λ₃), and thus can serve as a test for materials of this kind. Under certain constitutive assumptions, there always exists a cylindrical eversion deformation that satisfies exact pointwise traction free boundary conditions over the entire surface of the cylinder. For such solutions the cavity must remain open upon eversion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical-Analytic Solution of Boundary-Value Stress-Strain Problems for Rotating Anisotropic Cylinders

A numerical-analytical approach is proposed to solve boundary-value stress-strain problems for hollow inhomogeneous cylinders under centrifugal loading. Their elastic characteristics vary in both radial and circumferential directions. The governing system of ordinary differential equations is derived using Fourier series for stresses and displacements. It is solved by the discrete-orthogonalization method. Solutions to specific problems are exemplified __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 82–87, August 2005.     

同心旋转圆柱间粘弹性流的非线性动力学模型

基于小间隙假设,将速度场和应力场有杉Ｆｏｕｒｉｅｒ展开式截断近似,推导了同心旋转圆柱间Ｏｌｄｒｏｙｄ－Ｂ型流体的六维动力系统,探讨了高分子添加对滑动轴承间油膜稳定性的影响。结果表明,少量的高分子添加剂具有推迟流体层流失稳的作用。     

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.     

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.     

轴向冲击载荷作用下直杆弹性动态屈曲的研究

本文利用分叉的概念,从应力波理论出发,把轴向冲击载荷作用下直杆的弹性动态屈曲问题化为特征值问题,从而导出直杆动态屈曲的临界分叉准则。同时,本文对一端固支直杆受轴向冲击时的弹性屈曲问题进行了实验研究,测到的临界屈曲时间和理论预计值基本符合,证实了理论模型的合理性.理论和实验两方面的结果都表明,直杆受轴向冲击载荷作用时的屈曲现象发生在波动过程的前期,因此在处理这类问题时考虑波动效应是必要的。     

A Variational Approach to Obstacle Problems for Shearable Nonlinearly Elastic Rods

We use variational methods to study obstacle problems for geometrically exact (Cosserat) theories for the planar deformation of nonlinearly elastic rods. These rods can suffer flexure, extension, and shear. There is a marked difference between the behavior of a shearable and an unshearable rod. The set of admissible deformations is not convex, because of the exact geometry used. We first investigate the fundamental question of describing contact forces, which we necessarily treat as vector‐valued Borel measures. Moreover, we introduce techniques for describing point obstacles. Then we prove existence for a very large class of problems. Finally, using nonsmooth analysis for handling the obstacle, we show that the Euler‐Lagrange equations are satisfied almost everywhere. These equations provide very detailed structural information about the contact forces. Accepted June 3, 1996     

Internally Pressurized Radially Polarized Piezoelectric Cylinders

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary-value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distributions in the cylinder are obtained numerically for two typical piezoceramics of technological interest, namely PZT-4 and BaTiO₃. It is shown that the hoop stresses in a cylinder composed of these materials can be made virtually uniform throughout the cross-section by applying an appropriate set of boundary conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact Solutions and Material Tailoring for Functionally Graded Hollow Circular Cylinders

We employ the Airy stress function to derive analytical solutions for plane strain static deformations of a functionally graded (FG) hollow circular cylinder with Young’s modulus E and Poisson’s ratio v taken to be functions of the radius r. For E ₁ and v ₁ power law functions of r, and for E ₁ an exponential but v ₁ an affine function of r, we derive explicit expressions for stresses and displacements. Here E ₁ and v ₁ are effective Young’s modulus and Poisson’s ratio appearing in the stress-strain relations. It is found that when exponents of the power law variations of E ₁ and v ₁ are equal then stresses in the cylinder are independent of v ₁; however, displacements depend upon v ₁. We have investigated deformations of a FG hollow cylinder with the outer surface loaded by pressure that varies with the angular position of a point, of a thin cylinder with pressure on the inner surface varying with the angular position, and of a cut circular cylinder with equal and opposite tangential tractions applied at the cut surfaces. When v ₁ varies logarithmically through-the-thickness of a hollow cylinder, then the maximum radial stress, the maximum hoop stress and the maximum radial displacements are noticeably affected by values of v ₁. Conversely, we find how E ₁ and v ₁ ought to vary with r in order to achieve desired distributions of a linear combination of the radial and the hoop stresses. It is found that for the hoop stress to be constant in the cylinder, E ₁ and v ₁ must be affine functions of r. For the in-plane shear stress to be uniform through the cylinder thickness, E ₁ and v ₁ must be functions of r ². Exact solutions and optimal design parameters presented herein should serve as benchmarks for comparing approximate solutions derived through numerical algorithms.