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2017 | Buch

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Modeling Nonlinear Problems in the Mechanics of Strings and Rods

The Role of the Balance Laws

verfasst von: Oliver M. O'Reilly

Verlag: Springer International Publishing

Buchreihe : Interaction of Mechanics and Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

This book presents theories of deformable elastic strings and rods and their application to broad classes of problems. Readers will gain insights into the formulation and analysis of models for mechanical and biological systems. Emphasis is placed on how the balance laws interplay with constitutive relations to form a set of governing equations. For certain classes of problems, it is shown how a balance of material momentum can play a key role in forming the equations of motion. The first half of the book is devoted to the purely mechanical theory of a string and its applications. The second half of the book is devoted to rod theories, including Euler’s theory of the elastica, Kirchhoff ’s theory of an elastic rod, and a range of Cosserat rod theories. A variety of classic and recent applications of these rod theories are examined. Two supplemental chapters, the first on continuum mechanics of three-dimensional continua and the second on methods from variational calculus, are included to provide relevant background for students.

This book is suited for graduate-level courses on the dynamics of nonlinearly elastic rods and strings.

Inhaltsverzeichnis

Frontmatter

Mechanics of Strings

Frontmatter

Chapter 1. Mechanics of a String

Abstract

We start this book with arguably the simplest theory of a deformable elastic body - that of a string or, as it is also known, a one-dimensional continuum. Our motivation is the development of a theory that can accommodate a range of effects such as gravity, spatial discontinuities in velocity, applied forces which are concentrated at a point, large displacements and stretches, and nonlinear material behavior. This theory is used to develop models for a variety of problems ranging from chains to conveyor belts and bungee cords to hanging cables. Initially a wide range of kinematical results are established. Then, the balance laws are presented and a methodology for using these laws to establish the equations governing the motions of both inextensible and elastic strings is presented.

Oliver M. O’Reilly

Chapter 2. Applications of the Mechanics of a String

Abstract

We now turn to applications of the treatment in the previous chapter and develop mechanical models for the dynamics of a string. Many of the systems we consider in this chapter are classical but the analyses of their equations of motion is heavily influenced by treatments of, and novel insights into, these problems that have appeared during the past twenty years. Several of these problems involve chains which we model as a string in the presence of a gravitational body force. The chain fountain, a chain being dropped into a heap, and a chain with a fold are among the problems considered. The analysis of these systems serves to illuminate the roles played by momentum, energy, and material momentum in generating a closed system of equations to determine the dynamics of the string.

Oliver M. O’Reilly

Chapter 3. Link, Writhe, and Twist

Abstract

We review the concepts of writhe, twist, and linking as applied to space curves and ribbons. The application of these concepts to DNA is also discussed.

Oliver M. O’Reilly

Mechanics of Rods

Frontmatter

Chapter 4. Theory of the Elastica and a Selection of Its Applications

Abstract

The theory of the elastica is discussed in this chapter. In addition to classical buckling problems, several applications of this theory to rod-like bodies adhering to rigid substrates are discussed.

Oliver M. O’Reilly

Chapter 5. Kirchhoff’s Rod Theory

Abstract

The theory of an elastic rod whose centerline is inextensible and whose cross sections remain plane and normal to the centerline is discussed. This theory, which is known as Kirchhoff rod theory, is presented in the modern context of a Cosserat rod theory. The governing equations for this widely used theory result in a set of equations to determine a rotation tensor P and a position vector r. This theory has a celebrated history in part because of Kirchhoff’s discovery that the equations governing static deformations of the rod are analogous to those for the rotational motion of a rigid body. A range of applications of the theory is also presented in this chapter. These examples include a terminally loaded rod which is bent and twisted and an initially curved rod which is straightened.

Oliver M. O’Reilly

Chapter 6. Theory of an Elastic Rod with Extension and Shear

Abstract

We now consider a generalization of Kirchhoff’s rod theory to a theory which can accommodate extensibility of the centerline and transverse shear of the cross sections. The theory is constructed by allowing the rod to extend and shear. Consequently, the only significant change to the balance laws lies in the strain energy function. Theories of this type were developed in the early 1970s by various authors including Antman [10], Green and Laws [130], and Reissner [299, 300]. Antman’s papers discussing this theory have been hugely influential. This theory has several aspects which has made it popular in the applied mathematics, biophysics, computational mechanics, and mechanics communities (see, e.g., [12, 86, 158, 199, 327]). In addition to discussing applications of the theory to the stretching of DNA molecules, the linearized version of the theory is shown to include Timoshenko’s beam theory. We also take this opportunity to discuss a treatment of material symmetry for rods.

Oliver M. O’Reilly

Chapter 7. Green and Naghdi’s Rod Theory

Abstract

The rod theory discussed in this chapter originated in a paper by Green and Laws in 1966 [128]. In this theory, the material curve is extensible, and the directors d _α can change their length and relative orientation. This theory was further developed in a series of papers by Green, Naghdi, and several of their coworkers. They also showed how it could be established by integrating the three-dimensional equations of continuum mechanics. Here, the rod theory is presented in the most general form for an elastic rod with two directors and possible discontinuities. The linearized theory is discussed in a series of exercises and well-known rod theories that can be considered as constrained theories are presented.

Oliver M. O’Reilly

Background Material

Frontmatter

Chapter 8. A Rapid Review of Some Elements of Continuum Mechanics

Abstract

In this supplemental chapter, we review several elements of continuum mechanics. The main topics we cover are curvilinear coordinates, stress tensors, balance laws, constitutive equations, and kinematical constraints. The material covered is background needed for the development of rod and string theories from three-dimensional considerations that are presented throughout the text.

Oliver M. O’Reilly

Chapter 9. Variational Methods

Abstract

Several applications of methods from the calculus of variations to rod theories are presented in this chapter.

Oliver M. O’Reilly

Backmatter

Titel: Modeling Nonlinear Problems in the Mechanics of Strings and Rods
verfasst von: Oliver M. O'Reilly
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-50598-5
Print ISBN: 978-3-319-50596-1
DOI: https://doi.org/10.1007/978-3-319-50598-5

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