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2022 | OriginalPaper | Chapter

5. Generic Balances

Author : Paul Steinmann

Published in: Spatial and Material Forces in Nonlinear Continuum Mechanics

Publisher: Springer International Publishing

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Abstract

This chapter represents the formulation of generic balances for generic volume as well as surface and curve extensive quantities, thereby highlighting their global and local formats and resorting in both cases to material and spatial control volumes as well as control surfaces and control curves.

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previous chapter Kinematics at Singular Sets
next chapter Kinematical ‘Balances’*
Footnotes
1
The intrinsic flux \({\boldsymbol{Z}}\) is related to the material configuration and the material time derivative of the balanced quantity \({\boldsymbol{z}}_\mathrm{m}\). The dynamic flux \({\boldsymbol{Z}}+{\boldsymbol{z}}_\mathrm{m}\otimes {\boldsymbol{W}}\) may be considered as a (right-sided referential\(\rightarrow \)material) Piola transformation of a corresponding flux \(\pmb {\mathcal{{Z}}}\) as related to the reference configuration and the total time derivative of the balanced quantity \({\boldsymbol{z}}_\mathrm{r}\). The latter denotes a density per unit volume in the reference configuration. The referential perspective on continuum mechanics is elaborated in much more detail within Chaps. 9 and 10 on Virtual Work and the Variational Setting, respectively.
 
Literature
1.
Petryk H, Mroz Z (1986) Time derivatives of integrals and functionals defined on varying volume and surface domains. Arch Mech 38:697–724MathSciNetMATH
Metadata
Title
Generic Balances
Author
Paul Steinmann
Copyright Year
2022
Book
Spatial and Material Forces in Nonlinear Continuum Mechanics

Print ISBN: 978-3-030-89069-8

Electronic ISBN: 978-3-030-89070-4

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-030-89070-4_5

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