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1994 | Book

Introduction and Mathematical Preliminaries Read chapter Read first chapter

Introduction to Linear Elasticity

Author: Phillip L. Gould

Publisher: Springer New York

Included in: Professional Book Archive

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About this book

"Elasticity is one of the crowning achievements of Western culture!" ex­ claimed my usually reserved colleague Professor George Zahalak during a meeting to discuss the graduate program in Solid Mechanics. Although my thoughts on the theory of elasticity had not been expressed in such noble terms, it was the same admiration for the creative efforts of the premier physicists, mathematicians and mechanicians of the 19th and 20th centuries that led me to attempt to popularize the basis of solid mechanics in this introductory form. The book is intended to provide a thorough grounding in tensor-based theory of elasticity, which is rigorous in treatment but limited in scope. It is directed to advanced undergraduate and graduate students in civil, mechani­ calor aeronautical engineering who may ultimately pursue more applied studies. It is also hoped that a few may be inspired to delve deeper into the vast literature on the subject. A one-term course based on this material may replace traditional Advanced Strength of Materials in the curriculum, since many of the fundamental topics grouped under that title are treated here, while those computational techniques that have become obsolete due to the availability of superior, computer-based numerical methods are omitted. Little, if any, originality is claimed for this work other than the selection, organization and presentation of the material. The principal historical con­ tributors are noted in the text and several modern references are liberally cited.

Table of Contents

Frontmatter
Chapter 1. Introduction and Mathematical Preliminaries
Abstract
The theory of elasticity comprises a consistent set of equations which uniquely describe the state of stress, strain and displacement at each point within an elastic deformable body. Solutions of these equations fall into the realm of applied mathematics, while applications of such solutions are of engineering interest. When elasticity is selected as the basis for an engineering solution, a rigor is accepted that distinguishes this approach from the alternatives, which are mainly based on the strength of materials with its various specialized derivatives such as the theories of rods, beams, plates and shells. The distinguishing feature between the various alternative approaches and the theory of elasticity is the pointwise description embodied in elasticity, without resort to expedients such as Navier’s hypothesis of plane sections remaining plane.
Phillip L. Gould
Chapter 2. Traction, Stress and Equilibrium
Abstract
An approach to the solution of problems in solid mechanics is to establish relationships first between applied loads and internal stresses and, subsequently, to consider deformations. Another approach is to examine deformations initially, and then proceed to the stresses and applied loads. Regardless of the eventual solution path selected, it is necessary to derive the component relationships individually. In this chapter, the first set of equations, which describe equilibrium between external and internal forces and stresses, are derived.
Phillip L. Gould
Chapter 3. Deformations
Abstract
Displacements with respect to a reference coordinate system may be physically observed, calculated, or measured (at least on the surface) for a deformed elastic body. Each displacement may be considered to have two components, one of which is due to relative movements or distortions within the body and the other which is uniform throughout the body, so-called rigid body motion. The relationships between the displacements and the corresponding internal distortions are known as the kinematic or the strain-displacement equations of the theory of elasticity and may take several forms depending on the expected magnitude of the distortions and displacements.
Phillip L. Gould
Chapter 4. Material Behavior
Abstract
The need to connect the equilibrium equations derived in Chapter 2 with the kinematic relations developed in Chapter 3 was pointed out earlier. This coupling is accomplished by considering the mechanical properties of the materials for which the theory of elasticity is to be applied and is expressed by constitutive or material laws.
Phillip L. Gould
Chapter 5. Formulation, Uniqueness and Solution Strategies
Abstract
In the previous chapters, we have derived the essential equations of the linear theory of elasticity. Repeated here in terse form, the equilibrium conditions [Eqs. (2.34)] are
$$\sigma_{ij,i}+f_{j}=0;$$
(5.1)
The kinematic relations [Eqs. (3.14)] are
$$\varepsilon_{ij}={1\over 2}(u_{i,j}+u_{j,i}),$$
(5.2)
with the compatibility constraints [Eqs. (3.60)]
$$\varepsilon_{ij,kl}+\varepsilon_{kl,ij}=\varepsilon_{ik,jl}+\varepsilon_{jl,ik};$$
(5.3)
and the constitutive law [Eqs. (4.18) and (4.19)] is given by
$$\sigma_{ij}=\lambda\delta_{ij}\varepsilon_{kk}+2\mu\varepsilon_{ij}$$
(5.4)
or
$$\varepsilon_{ij}={1\over E}\left[(1+v)\sigma_{ij}-v\delta_{ij}\sigma_{kk}\right].$$
(5.5)
Phillip L. Gould
Chapter 6. Extension, Bending and Torsion
Abstract
It is instructive to examine some familiar problems, readily solved by elementary theories based on a strength of materials approach, using the theory of elasticity. We anticipate that the elementary solutions are approximately correct, but deficient or incomplete in some way. In each case, the isotropic material law is assumed to be applicable.
Phillip L. Gould
Chapter 7. Two-Dimensional Elasticity
Abstract
We briefly mentioned in Section 2.6.2 that an elasticity problem may be reduced from three- to two-dimensions if there is no traction on one plane passing through the body. This state is known as plane stress since all nonzero stresses are confined to planes parallel to the traction-free plane.
Phillip L. Gould
Chapter 8. Bending of Thin Plates
Abstract
In the preceding chapter, a thin elastic plate loaded in the plane of the plate was analyzed. The resulting deformations are confined to the plane of the plate in accordance with the plane strain assumption.
Phillip L. Gould
Chapter 9. Time-Dependent Effects
Abstract
The strains in an elastic body may be computed from a specified displacement field using the equations of compatibility, regardless of whether the displacements arise from static or dynamic excitation. The corresponding stresses and, indeed, the displacements themselves may be dependent on the rate characteristics of the loading function. Therefore, the time derivatives of the displacements, i.e., velocities and accelerations, enter into these equations.
Phillip L. Gould
Chapter 10. Energy Principles
Abstract
The theory of elasticity may be developed from energy considerations, leading to the field equations in the form of differential equations. This approach does not promise any computational advantage from the standpoint of analytical solutions, since the same equations found from the classical formulations are produced. However, for the pursuit of numerical solutions, energy methods are extensively developed and are the basis of powerful contemporary programs for solving complex problems in solid mechanics. This development is rather recent and was brought about by the use of the digital computer. This probably explains the relatively few applications of energy methods to elasticity problems found in the classical texts. However, even in the 18th century, the mathematician Leonhard Euler contrasted the direct formulation of the governing differential equations, known then as the method of effective causes, with the energy approach, known then as the method of final causes, in an argument that skillfully blended science and theology (this may have been diplomatic considering the well-known plight of perhaps the first “elastician,” Galileo, a century earlier). Euler wrote [10.1]
Phillip L. Gould
Chapter 11. Strength and Failure Criteria
Abstract
The application of the rigorous methods of analysis embodied in the theory of elasticity is naturally of interest to engineers. While a broad discussion of this issue is beyond our scope, it is of interest to introduce the fundamental basis of such application, namely, the comparison of the analytical results obtained from an elasticity solution to the expected capacity of the resisting material.
Phillip L. Gould
Chapter 12. Something New
Abstract
It was stated in the first introductory Section 1.1 that the theory of elasticity originated in the first half of the 19th century. Most of the original work supporting the presentation in this introductory text is many decades old.
Phillip L. Gould
Backmatter
Metadata
Title
Introduction to Linear Elasticity
Author
Phillip L. Gould
Copyright Year
1994
Publisher
Springer New York
Electronic ISBN
978-1-4612-4296-3
Print ISBN
978-1-4612-8728-5
DOI
https://doi.org/10.1007/978-1-4612-4296-3