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2016 | OriginalPaper | Chapter

4. Three-Dimensional Continuum Mechanics

Authors : Konstantin Naumenko, Holm Altenbach

Published in: Modeling High Temperature Materials Behavior for Structural Analysis

Publisher: Springer International Publishing

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Abstract

The objective of continuum mechanics is to develop mathematical models to analyze the behavior of idealized three-dimensional bodies.

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previous chapter Elementary Uni-axial Constitutive Models
next chapter Constitutive Models
Footnotes
1
In many cases strains can be infinitesimal, but rotations finite. One example is a thin plate strip which can be bent into a ring such that the strains remain infinitesimal but cross section rotations are large.
 
2
The tensor \(\pmb D\) is in general not a time derivative of a strain tensor.
 
3
This can be verified applying the integral theorem (B.3.4)\(_1\) with \(\varphi =1\).
 
4
In some books of continuum mechanics and applied mathematics the stress tensor is defined as \({\large \pmb \sigma } =\pmb \sigma _{(\pmb e_1)}\otimes \pmb e_1+\pmb \sigma _{(\pmb e_2)}\otimes \pmb e_2+\pmb \sigma _{(\pmb e_3)}\otimes \pmb e_3\) such that the Cauchy formula is \(\pmb \sigma _{(\pmb n)}= {\large \pmb \sigma }\pmb {\cdot }\pmb n\). Formally this definition differs from (4.3.89) by transpose. It might be more convenient, as it is closer to the matrix algebra. For engineers dealing with internal forces it is more natural to use (4.3.89). Indeed, to analyze a stress state we need to cut the body first and to specify the normal to the cut plane. Only after that we can introduce the internal force. The sequence of these operations is clearly seen in (4.3.89).
 
5
With regard to structural analysis applications discussed in this book it is enough to identify the angular momentum as the moment of momentum and the resultant moment as the moment of forces. In contrast, within the micropolar theories material points are equipped by tensor of inertia. The resultant moment includes surface and body moments which are not related to moment of forces, e.g. Altenbach et al. (2003), Eringen (1999), Nowacki (1986).
 
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Metadata
Title
Three-Dimensional Continuum Mechanics
Authors
Konstantin Naumenko
Holm Altenbach
Copyright Year
2016
Book
Modeling High Temperature Materials Behavior for Structural Analysis

Print ISBN: 978-3-319-31627-7

Electronic ISBN: 978-3-319-31629-1

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-31629-1_4

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