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Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

2. Newtonian and Eshelbian Mechanics

verfasst von : Marcus Olavi Rüter

Erschienen in: Error Estimates for Advanced Galerkin Methods

Verlag: Springer International Publishing

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Abstract

The objective of this chapter is to present an introduction to the theory of continuum mechanics of elastic structures. Classical continuum mechanics deals with finding the spatial configuration of an elastic body that is subjected to external forces. This forward problem is attributed to Sir Isaac Newton and therefore termed Newtonian mechanics. In the associated inverse problem, which is attributed to John Douglas Eshelby and therefore termed Eshelbian mechanics, we are concerned with the forces applied to the spatial configuration. In Newtonian mechanics, the applied forces are of a physical nature, and as a result, the associated stress that arises in the spatial configuration of the elastic body is well known as the (physical) Cauchy stress. In Eshelbian mechanics, on the other hand, the deformed elastic body is subjected to so-called material forces, and the resulting stress in the initial configuration (of the forward problem) is termed the (material) Eshelby stress. In this chapter, the stress tensors that naturally appear in both Newtonian and Eshelbian mechanics are systematically derived for both compressible and (nearly) incompressible materials.

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Vorheriges Kapitel Introduction
Nächstes Kapitel Boundary Value Problems
Fußnoten
1
Note that it is conventional practice to denote the divergence operator equivalently as \(\nabla _{x} \cdot (\cdot )\). However, this is not always possible, as we shall see in Sect. 2.2.4. Moreover, this notation is not consistent with the (spatial) gradient operator \((\cdot ) \otimes \nabla _{x}\).
 
2
Note that it is customary in the literature to express the material tractions \({\varvec{T}}_{\text {mat}}\) by \({\varvec{\Sigma }}^{\sharp } \cdot {\varvec{N}}\), according to the stress theorem (2.24).
 
3
To be consistent with the tensor gradient operators \((\cdot ) \otimes \nabla _{x}\) and \((\cdot ) \otimes \nabla _{X}\), we use the notations \((\cdot ) \nabla _{x}\) and \((\cdot ) \nabla _{X}\) in the vector case.
 
4
Note that the material part \(\mathbb {A}_{\text {mat}}\) is not related to material forces in Eshelbian mechanics.
 
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Metadaten
Titel
Newtonian and Eshelbian Mechanics
verfasst von
Marcus Olavi Rüter
Copyright-Jahr
2019
Buch
Error Estimates for Advanced Galerkin Methods

Print ISBN: 978-3-030-06172-2

Electronic ISBN: 978-3-030-06173-9

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-06173-9_2

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