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Vector, Tensors, and Related Matters Read chapter Read first chapter

2020 | OriginalPaper | Chapter

5. Linear Elasticity: General Considerations and Boundary-Value Problems

Author : Ciprian D. Coman

Published in: Continuum Mechanics and Linear Elasticity

Publisher: Springer Netherlands

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Abstract

In this chapter, we introduce the mathematical model for a linearly elastic solid and its associated boundary-value problems. This is achieved by a judicious particularisation of the general theory developed in the previous chapters; broadly speaking, the reference and current configurations will be assumed to be very close to each other (in a sense that will be made clear shortly). The upshot of this simplification is the linearity of the aforementioned boundary-value problems, which can then be solved by a number of indirect strategies involving: superposition, semi-inverse approaches, and the Saint-Venant’s Principle. The last section touches upon some well-established approximations whereby a three-dimensional situation is reduced to a two-dimensional problem. These specific approximations are taken up in much greater detail in some of the subsequent chapters.

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previous chapter Constitutive Relationships
next chapter Compatibility of the Infinitesimal Deformation Tensor
Footnotes
1
The ‘big Oh’ notation is widely used in Mathematics: if f(x) and g(x) are two functions of x, then \(f(x)=\mathscr {O}(g(x))\) as \(x\rightarrow x_0\) if there exist \(\delta ,\,M>0\) such that \(|f(x)|\le M|g(x)|\) if \(0<|x - x_0|<\delta \).
 
2
The expression on the right-hand side of (5.19) is a particular type of fourth-order isotropic tensor; see the discussion in Sect. 1.​11.
 
3
The general problem of determining the displacement field from the strain tensor will be taken up in greater detail in the next chapter. In this example the integration of the kinematic equations is elementary; for details of a similar calculation see Example 6.3.
 
4
These are sometimes referred to as fibres.
 
Literature
1.
Barber JR (2002) Elasticity. Kluwer Academic Publishers, Dordrecht, The Netherlands
2.
Biezeno CB, Grammel R (1955) Engineering dynamics, vol 1 (Theory of Elasticity). Blackie & Son Ltd., London and Glasgow
3.
Biezeno CB, Grammel R (1956) Engineering dynamics, vol 2 (Elastic Problems of Single Machine Elements). Blackie & Lon Ltd., London and Glasgow
4.
Chou PC, Pagano NJ (1967) Elasticity: tensor, dyadic, and engineering approaches. D. Van Nostrand Company Inc, Princeton, New Jersey
5.
Filonenko-Borodich M (1963) Theory of elasticity. Mir Publishers, Moscow
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Leipholz H (1974) Theory of elasticity. Noordhoff International Publishing, Leyden
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Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. Mir Publishers, Moscow
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Little RW (1973) Elasticity. Prentice-Hall Inc, Englewood Cliffs, New Jersey
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Love AEH (2011) A treatise of the mathematical theory of elasticity. Dover Publications, Mineola, New York
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Malvern LE (1969) Introduction to the mechanics of a continuum medium. Prentice-Hall Inc, Englewood Cliffs, New Jersey
11.
Nadeau G (1964) Introduction to elasticity. Holt, Rinehart and Winston Inc, New York
12.
Teodorescu PP (2013) Treatise on classical elasticity. Springer, Dordrecht, The NetherlandsCrossRef
13.
Timoshenko SP, Goodier JN (1970) Theory of elasticity, International edn. McGraw-Hill Book Company, Auckland
Metadata
Title
Linear Elasticity: General Considerations and Boundary-Value Problems
Author
Ciprian D. Coman
Copyright Year
2020
Publisher
Springer Netherlands
Book
Continuum Mechanics and Linear Elasticity

Print ISBN: 978-94-024-1769-2

Electronic ISBN: 978-94-024-1771-5

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-94-024-1771-5_5

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