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Cosserat Theories: Shells, Rods and Points

verfasst von: M. B. Rubin

Verlag: Springer Netherlands

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

Continuum Continuum mechanics mechanics provides provides a a theoretical theoretical structure structure for for analyzing analyzing the the response response of of materials materials to to mechanical mechanical and and thermal thermal loads. loads. One One of of the the beauties beauties of of continuum continuum mechanics mechanics is is that that the the fundamental fundamental balance balance laws laws (conservation (conservation of of mass mass and and balances balances of of linear linear momentum, momentum, angular angular momentum, momentum, energy energy and and entropy) entropy) are are valid valid for for all all simple simple materials. materials. Most Most of of the the modern modern research research in in continuum continuum mechanics mechanics focuses focuses on on the the development development of of constitutive constitutive equations equations which which are are used used to to characterize characterize the the response response of of a a particular particular class class of of materials materials (e.g. (e.g. invisicid invisicid fluids, fluids, viscous viscous fluids, fluids, elastic elastic solids, solids, viscoelastic viscoelastic solids, solids, elastic elastic plastic solids, elastic-viscoplastic solids, etc.). plastic solids, elastic-viscoplastic solids, etc.).

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In the usual simple three-dimensional continuum it is sufficient to describe the motion of the continuum by the position vector which identifies the location of each material point as a function of time. For the purely mechanical theory, the laws of conservation of mass and the balance of linear momentum are used to determine the present values of the mass density and this position vector. Also, the balance of angular momentum is used to place restrictions on the constitutive equations of the continuum (i.e. the symmetry of the stress tensor).

M. B. Rubin

Chapter 2. Basic Tensor Operations in Curvilinear Coordinates

Abstract

In continuum mechanics as well as in other branches of engineering and physics, it is necessary to develop mathematical models that describe phenomena observed in the physical world. Almost always it is necessary to describe the location of a material point in space relative to some fixed point and relative to some specified fixed axes. The specific choice of these axes remains arbitrary but it is usually guided by desire to simplify some aspect of the description of the material response. Obviously, the actual response of a given material to a specific loading must be independent of the particular choice of these fixed axes. Consequently, it is necessary to use mathematical tools that automatically ensure that predictions of the resulting mathematical equations are independent of the specific choice of these axes. Vectors and tensors are such mathematical tools.

M. B. Rubin

Chapter 3. Three-Dimensional Continua

Abstract

This chapter provides a brief review of the basic equations that describe the motion of a three-dimensional continua.

M. B. Rubin

Chapter 4. Cosserat Shells

Abstract

A shell-like structure or shell is a three-dimensional body that has special geometric features. Most importantly, the shell is a three-dimensional body that is considered to be “thin” in one of its dimensions (see Fig. 4.1.1). In particular, the shell is characterized by its major surfaces (bottom and top) and its lateral surface. From another point of view, the shell is considered to be a material surface S which has some finite thickness bounded by the major surfaces. If this surface S is flat, then the shell-like structure is called a plate, otherwise it is called a shell. Such shell-like structures appear in practice in many applications. For example, the floors, walls and roofs of many buildings are flat surfaces that can be modeled as plates, whereas the surfaces of an airplane body or car body, the human skull and the veins and arteries in the human circulatory system are curved surfaces that can be modeled as shells.

M. B. Rubin

Chapter 5. Cosserat Rods

Abstract

A rod-like structure, or rod, is a three-dimensional body that has special geometric features. Most importantly, the rod is a three-dimensional body that is considered to be “thin” in two of its dimensions (see Fig. 5.1.1). In particular, the rod is characterized by its ends and its lateral surface. From another point of view, the rod is considered to be a material curve C which has some finite thickness bounded by the rod’s lateral surface. If this curve C is a straight line, then the rod-like structure is called a beam, otherwise it is called a rod. Such rod-like structures appear in practice in many applications. For example, the main supporting structures in buildings, and the connecting bars in trusses can be modeled as beams, whereas the curved reinforcement ribs of airplane wings and submarines, and the double helix of DNA molecules can be modeled as rods.

M. B. Rubin

Chapter 6. Cosserat Points

Abstract

A point-like structure, or Cosserat point, is a three-dimensional body that has special geometric features. Most importantly, the Cosserat point is a three-dimensional body that is considered to be “thin” in all three of its dimensions. Therefore, it is essentially a zerodimensional point surrounded by some finite but small region of material. Sometimes this region P* is bounded by a smooth closed surface ∂P*, while other times it is a polyhedron bounded by S planar surfaces ∂P_J* (J=1,2,...,S) (see Fig. 6.1.1). In all cases, the Cosserat point theory provides a simple continuum description of the motion and deformation of this small point-like structure.

M. B. Rubin

Chapter 7. Numerical Solutions Using Cosserat Theories

Abstract

Numerical solution procedures for problems in continuum mechanics using finite element methods are very well developed and are used in many commercial computer codes for engineering design and analysis. A number of books on finite elements describe both the engineering and mathematical aspects of the finite element method (e. g. Strang and Fix, 1973; Huebner, 1975; Desai, 1979; Becker et al, 1981; Bathe, 1982; Carey and Oden, 1983 and 1984; Reddy, 1985; Hughes, 1987; and Zienkiewicz and Taylor, 1989 and 1991). In its simplest form the finite element method determines an approximate solution of a set of partial differential equations by first representing these equations in a weak form. This weak form can be developed by the method of weighted residuals or can be expressed in a functional representation when one exits. Then, the independent variables are expressed in terms of shape functions (over space and/or time), and a set of ordinary differential equations in time, or algebraic equations, are obtained for the coefficients of these shape functions. Much research focuses on developing efficient methods for solving the resulting large numbers of nonlinear algebraic equations and on determining special shape functions and element formulations which have numerically desirable properties.

M. B. Rubin

Backmatter

Titel: Cosserat Theories: Shells, Rods and Points
verfasst von: M. B. Rubin
Verlag: Springer Netherlands
Electronic ISBN: 978-94-015-9379-3
Print ISBN: 978-90-481-5531-6
DOI: https://doi.org/10.1007/978-94-015-9379-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. Basic Tensor Operations in Curvilinear Coordinates

Chapter 3. Three-Dimensional Continua

Chapter 4. Cosserat Shells

Chapter 5. Cosserat Rods

Chapter 6. Cosserat Points

Chapter 7. Numerical Solutions Using Cosserat Theories

Backmatter