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Published in:
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2022 | OriginalPaper | Chapter

10. Variational Setting

Author : Paul Steinmann

Published in: Spatial and Material Forces in Nonlinear Continuum Mechanics

Publisher: Springer International Publishing

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Abstract

This chapter expands on the variational setting in terms of extended Hamilton and Dirichlet principles for conservative elasto-dynamic and elasto-static cases, respectively, and carefully analyzes the resulting spatial and material Euler-Lagrange equations.

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Footnotes
1
A step-by-step derivation is as follows:
$$\begin{aligned}&\widehat{\text{ Div }}(\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}})= \big [[\widehat{\boldsymbol{ G}}{}^\alpha \otimes \widehat{\boldsymbol{ g}}_ \alpha ]\cdot \widehat{\boldsymbol{ P}}\big ]_{,\beta }\cdot \widehat{\boldsymbol{ G}}{}^\beta \\= & {} [\widehat{\boldsymbol{ G}}{}^\alpha \otimes \widehat{\boldsymbol{ g}}_ \alpha ]\cdot \widehat{\boldsymbol{ P}} _{,\beta }\cdot \widehat{\boldsymbol{ G}}{}^\beta + [\widehat{\boldsymbol{ G}}{}^\alpha \otimes \widehat{\boldsymbol{ g}}_{\alpha ,\beta } + \widehat{\boldsymbol{ G}}{}^\alpha _{\phantom {\alpha },\beta }\otimes \widehat{\boldsymbol{ g}}_ \alpha ]\cdot \widehat{\boldsymbol{ P}} \cdot \widehat{\boldsymbol{ G}}{}^\beta \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [\widehat{\boldsymbol{ g}}_{\alpha ,\beta } \otimes \widehat{\boldsymbol{ G}}{}^\beta \otimes \widehat{\boldsymbol{ G}}{}^\alpha + \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta \otimes \widehat{\boldsymbol{ G}}{}^\alpha _{\phantom {\alpha },\beta }\big ] \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [\widehat{\boldsymbol{ g}}_{\alpha ,\beta } \otimes \widehat{\boldsymbol{ G}}{}^\alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta + \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta \otimes [-\widehat{\varGamma }{}^\alpha _{\phantom {\alpha }\beta \gamma }\,\widehat{\boldsymbol{ G}}{}^\gamma +\widehat{C}{}^\alpha _{\phantom {\alpha }\beta }\,\boldsymbol{ N}]\big ] \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [[\widehat{\boldsymbol{ g}}_{\alpha ,\gamma }\otimes \widehat{\boldsymbol{ G}}{}^\alpha -\widehat{\varGamma }{}^\alpha _{\phantom {\alpha }\beta \gamma }\, \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta ]\otimes \widehat{\boldsymbol{ G}}{}^\gamma +\widehat{C}{}^\alpha _{\phantom {\alpha }\beta }\, \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta \otimes \boldsymbol{ N} \big ] \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [[\widehat{\boldsymbol{ g}}_{\alpha ,\gamma }\otimes \widehat{\boldsymbol{ G}}{}^\alpha -\widehat{\varGamma }{}^\alpha _{\phantom {\alpha }\gamma \beta }\, \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta ]\otimes \widehat{\boldsymbol{ G}}{}^\gamma +\widehat{C}{}^\alpha _{\phantom {\alpha }\beta }\, \widehat{\boldsymbol{ F}}\cdot \widehat{\boldsymbol{ G}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta \otimes \boldsymbol{ N} \big ] \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [[\widehat{\boldsymbol{ g}}_{\alpha ,\gamma }\otimes \widehat{\boldsymbol{ G}}{}^\alpha -\widehat{\varGamma }{}^\alpha _{\phantom {\alpha }\gamma \beta }\, \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\beta +\widehat{C}{}^\alpha _{\phantom {\alpha }\gamma }\, \widehat{\boldsymbol{ g}}_ \alpha \otimes \boldsymbol{ N} ]\otimes \widehat{\boldsymbol{ G}}{}^\gamma + \widehat{\boldsymbol{ F}}\cdot \widehat{\boldsymbol{ C}} \otimes \boldsymbol{ N} \big ] \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:\big [\widehat{\boldsymbol{ g}}_{\alpha ,\gamma }\otimes \widehat{\boldsymbol{ G}}{}^\alpha + \widehat{\boldsymbol{ g}}_ \alpha \otimes [-\widehat{\varGamma }{}^\alpha _{\phantom {\alpha }\gamma \beta }\,\widehat{\boldsymbol{ G}}{}^\beta +\widehat{C}{}^\alpha _{\phantom {\alpha }\gamma }\,\boldsymbol{ N}]\big ]\otimes \widehat{\boldsymbol{ G}}{}^\gamma + [\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}}]:\widehat{\boldsymbol{ C}}\,\boldsymbol{ N} \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+ \widehat{\boldsymbol{ P}}:[\widehat{\boldsymbol{ g}}_{\alpha ,\gamma }\otimes \widehat{\boldsymbol{ G}}{}^\alpha + \widehat{\boldsymbol{ g}}_ \alpha \otimes \widehat{\boldsymbol{ G}}{}^\alpha _{\phantom {\alpha },\gamma }]\otimes \widehat{\boldsymbol{ G}}{}^\gamma + [\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}}]:\widehat{\boldsymbol{ C}}\,\boldsymbol{ N} \\= & {} \widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+\widehat{\boldsymbol{ P}}:\widehat{\nabla }_{\widehat{X}}\widehat{\boldsymbol{ F}}+ [\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}}]:\widehat{\boldsymbol{ C}}\,\boldsymbol{ N}. \end{aligned}$$
.
 
2
From the established result
$$ \widehat{\text{ Div }}(\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}})=\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+\widehat{\boldsymbol{ P}}:\widehat{\nabla }_{\widehat{X}}\widehat{\boldsymbol{ F}}+[\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}}]:\widehat{\boldsymbol{ C}}\,\boldsymbol{ N} $$
it follows for its tangential and normal parts, respectively, that
$$ \widehat{\boldsymbol{ I}}\cdot \widehat{\text{ Div }}(\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}})=\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ P}}+\widehat{\boldsymbol{ P}}:\widehat{\nabla }_{\widehat{X}}\widehat{\boldsymbol{ F}}\quad \text{ and } \quad \boldsymbol{ N}\cdot \widehat{\text{ Div }}(\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}})=[\widehat{\boldsymbol{ F}}{}^t\cdot \widehat{\boldsymbol{ P}}]:\widehat{\boldsymbol{ C}}. $$
.
 
3
A step-by-step derivation is as follows:
$$\begin{aligned} \boldsymbol{ N}\cdot \widehat{\text{ Div }}\widehat{\boldsymbol{ \varSigma }}{}'=\boldsymbol{ N}\cdot \widehat{\boldsymbol{ \varSigma }}{}'_{,\alpha }\cdot \widehat{\boldsymbol{ A}}{}^\alpha= & {} -\boldsymbol{ N}_{,\alpha }\cdot \widehat{\boldsymbol{ \varSigma }}{}'\cdot \widehat{\boldsymbol{ A}}{}^\alpha +\big [\boldsymbol{ N}\cdot \widehat{\boldsymbol{ \varSigma }}{}'\big ]_{,\alpha }\cdot \widehat{\boldsymbol{ A}}{}^\alpha \\= & {} -\big [\boldsymbol{ N}_{,\alpha }\otimes \widehat{\boldsymbol{ A}}{}^\alpha \big ]:\widehat{\boldsymbol{ \varSigma }}{}'+\widehat{\text{ Div }}\big (\boldsymbol{ N}\cdot \widehat{\boldsymbol{ \varSigma }}{}'\big )\\= & {} -\widehat{\nabla }_{\widehat{X}}\boldsymbol{ N}:\widehat{\boldsymbol{ \varSigma }}{}'=\widehat{\boldsymbol{ C}}:\widehat{\boldsymbol{ \varSigma }}{}'. \end{aligned}$$
.
 
Metadata
Title
Variational Setting
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_10

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