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2002 | Buch

Review of Mechanics of Materials Kapitel lesen Erstes Kapitel lesen

The Linearized Theory of Elasticity

verfasst von: William S. Slaughter

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

This book is derived from notes used in teaching a first-year graduate-level course in elasticity in the Department of Mechanical Engineering at the University of Pittsburgh. This is a modern treatment of the linearized theory of elasticity, which is presented as a specialization of the general theory of continuum mechanics. It includes a comprehensive introduction to tensor analysis, a rigorous development of the governing field equations with an emphasis on recognizing the assumptions and approximations in­ herent in the linearized theory, specification of boundary conditions, and a survey of solution methods for important classes of problems. Two- and three-dimensional problems, torsion of noncircular cylinders, variational methods, and complex variable methods are covered. This book is intended as the text for a first-year graduate course in me­ chanical or civil engineering. Sufficient depth is provided such that the text can be used without a prerequisite course in continuum mechanics, and the material is presented in such a way as to prepare students for subsequent courses in nonlinear elasticity, inelasticity, and fracture mechanics. Alter­ natively, for a course that is preceded by a course in continuum mechanics, there is enough additional content for a full semester of linearized elasticity.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Review of Mechanics of Materials
Abstract
The purpose of this chapter is to reacquaint the reader with some of the fundamental concepts of mechanics of materials, as it is taught to undergraduate engineering students. Mechanics of materials (sometimes referred to as “strength of materials”) is distinguished from the theory of elasticity by its relative disregard for mathematical rigor. In the usual undergraduate treatment of the subject, so-called “engineering” approximations and assumptions are casually proposed and accepted—their validity is based on experimental verification. For a great many engineering applications, this is perfectly acceptable. As long as one does not venture beyond the boundaries of common experience in utilizing the results from mechanics of materials, their accuracy can be relied upon.
William S. Slaughter
Chapter 2. Mathematical Preliminaries
Abstract
The principal objective of this chapter is to introduce the theory of tensor analysis. Other mathematical topics, such as the calculus of variations and the theory of functions of a complex variable, will be reviewed in subsequent chapters as they are needed. Three-dimensional Euclidean vector and point spaces are implicitly assumed throughout.
William S. Slaughter
Chapter 3. Kinematics
Abstract
A body is composed of discrete material quanta in the form of atoms, molecules, grains, polymer chains, and so forth. It is the central premise of continuum mechanics that this discrete nature of material can be neglected in studying the deformation of a body, provided the length scales of interest are large relative to the length scale of the material quanta. A body in this case is modeled as a continuum in which physical quantities distributed over the body are represented by fields that are continuous (except possibly at a finite number of surfaces, lines, and/or points).
William S. Slaughter
Chapter 4. Forces and Stress
Abstract
Material volumes interact by exerting forces on one another and it is through the actions of these forces that motion of a material volume occurs. Forces can be categorized either as body forces that act at a distance, such as the gravitational forces of attraction that exist between any two material volumes, or as surface forces that two material volumes exert on one another when they are in contact. An important concept in the analysis of forces is that of the free body and the identification of forces as either internal or external relative to a free body. Any material volume can be identified as a free body. Then, any forces exerted by one portion of the free body on another portion of the free body are said to be internal relative to the free body. Forces exerted on the free body by other material volumes that are not part of the free body are external relative to the free body. Whether a force is internal or external will depend on what material volume one chooses to be the free body. Whereas external body forces act, in general, on all the material points of a free body, external surface forces only act on material points on the free body’s boundary.
William S. Slaughter
Chapter 5. Constitutive Equations
Abstract
The equations of motion, given for instance in a referential formulation by (4.1.33) and (4.1.38), are generally insufficient to determine the stress field in a body. There are nine scalar components of stress and only six scalar equations of motion. The equations of motion do not directly relate the deformation of a body to the external forces applied to the body either. What is lacking are a statement of the particular boundary conditions of a problem (discussed in Chapter 6) and a consideration of the mechanical properties associated with the particular material that constitutes the body, relating deformation and stress in the form of constitutive equations.
William S. Slaughter
Chapter 6. Linearized Elasticity Problems
Abstract
Problems in the linearized theory of elasticity will require finding the solution of a set of coupled partial differential equations known as field equations, over a specified domain that is the region occupied by the reference configuration of the body, subject to prescribed boundary conditions on the boundary of the domain. In other words, linearized elasticity problems are boundary value problems.
William S. Slaughter
Chapter 7. Two-Dimensional Problems
Abstract
There is a class of problems in elasticity which, due to geometry and boundary conditions, have solution stress, strain, and displacement fields that are independent of one of the coordinate variables. The equations of elasticity for this class of problem reduce to a simplified form of the general equations. Recognizing that a body with given boundary conditions is a member of this class, in advance of solving the attendant boundary value problem, enables one to take advantage of these simplified equations. Members of this class of problem are subdivided into three types known as antiplane strain, plane strain, and plane stress, as discussed below. It will be seen that the reduced form of the equations of elasticity for the later two are functionally equivalent.
William S. Slaughter
Chapter 8. Torsion of Noncircular Cylinders
Abstract
Consider a cylindrical body of length L, the ends of which are subject to distributions of traction that are statically equivalent to equal and opposite torques ±T = ±Tê3. The lateral surface of the cylinder is traction-free. It is assumed that there is an axis that passes through the center of twist of each cross section of the cylinder (i.e., an axis on which the displacement perpendicular to the axis is zero) and this is defined to be the X3-axis (see Figure 8.1). Let δ be the cross section of the cylinder and let ∂δ be its boundary. The cross section δ is, in general, noncircular.
William S. Slaughter
Chapter 9. Three-Dimensional Problems
Abstract
Although the solution to some simple three-dimensional problems in linearized elasticity have been presented (e.g., spherical pressure vessels), the focus so far has been on special classes of problems for which the equations of elasticity could be reduced to a two-dimensional form. Some additional three-dimensional problems of interest will now be considered.
William S. Slaughter
Chapter 10. Variational Methods
Abstract
Variational methods involve the use of minimization principles such as the principles of minimum potential energy and minimum complementary energy. These minimization principles can be used to derive governing equations and boundary conditions for specialized classes of problems in elasticity. Another important application of energy methods is in finding approximate solutions to elasticity problems, which involves making assumptions about the relative accuracy of different approximations based on the minimization of certain quantities to be defined. For instance, this is the basis of the numerical algorithm known as the finite element method.
William S. Slaughter
Chapter 11. Complex Variable Methods
Abstract
There are certain properties of complex-valued functions of a complex variable, discussed below, that prove to be very useful in the study of two-dimensional potential theory with applications to heat flow, inviscid fluids, and linearized elasticity, among others. The focus here is on the application of complex variable methods to antiplane strain and plane strain/stress problems in linearized elasticity.
William S. Slaughter
Backmatter
Metadaten
Titel
The Linearized Theory of Elasticity
verfasst von
William S. Slaughter
Copyright-Jahr
2002
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0093-2
Print ISBN
978-1-4612-6608-2
DOI
https://doi.org/10.1007/978-1-4612-0093-2