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2022 | OriginalPaper | Chapter

12. Consequences of Thermodynamical 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 exploits the consequences of the thermodynamical balances and the resulting formats of the dissipation power inequalities by identifying the forces driving material (configurational) changes on the boundary and at singular surfaces as the appropriate contributions to the balance of material momentum. It is the here advocated thermodynamical derivation of material forces that qualifies the current approach as being dissipation-consistent.

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previous chapter Thermodynamical Balances
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Footnotes
1
Spatial objectivity is here particularized as invariance of the free energy density \(\psi _\mathrm{m}\) under superposed rigid body motions (srbm). These follow as
$$ \boldsymbol{y}(\boldsymbol{X}, t)\quad \rightarrow \quad \boldsymbol{Q}(t)\cdot \boldsymbol{y}(\boldsymbol{X}, t)+\boldsymbol{c}(t) $$
with \(\boldsymbol{Q}\in \text{ SO(3) }\) and \(\boldsymbol{c}\in \text{ T(3) }\) arbitrary elements from the special orthogonal and the translational (three-dimensional) groups, respectively. Thus, spatial objectivity requires
$$ \psi _\mathrm{m}(\boldsymbol{y}, \cdots )\doteq \psi _\mathrm{m}(\boldsymbol{Q}\cdot \boldsymbol{y}+\boldsymbol{c}, \cdots )\quad \forall \boldsymbol{Q}\in \text{ SO(3) }\quad \text{ and } \quad \forall \boldsymbol{c}\in \text{ T(3) }. $$
Obviously, this can only be satisfied for \(\psi _\mathrm{m}\not =\psi _\mathrm{m}(\boldsymbol{y})\). It is also recalled that spatial objectivity results in the dependence of \(\psi _\mathrm{m}\) on \(\boldsymbol{F}\) through \(\boldsymbol{C}\) (under srbm \(\boldsymbol{F}\rightarrow \boldsymbol{Q}\cdot \boldsymbol{F}\), whereas \(\boldsymbol{C}\) remains invariant, thus \(\psi _\mathrm{m}(\boldsymbol{C})\) remains invariant too) and thus, in agreement with the balance of spatial angular momentum, eventually also in the symmetry of the spatial description Kirchhoff stress \(\boldsymbol{\tau }=\boldsymbol{\tau }^t\).
 
Literature
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Javili A, McBride A, Steinmann P (2013) Numerical modelling of thermomechanical solids with highly conductive energetic interfaces. Int J Numer Methods Eng 93:551–574MathSciNetCrossRef
2.
Esmaeili A, Javili A, Steinmann P (2017) Highly-conductive energetic coherent interfaces subject to in-plane degradation. Math Mech Solids 22:1696–1716MathSciNetCrossRef
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Javili A, McBride A, Steinmann P (2012) Numerical modelling of thermomechanical solids with mechanically energetic (generalised) kapitza interfaces. Comput Mat Sci 65:542–551CrossRef
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Kaessmair S, Javili A, Steinmann P (2014) Thermomechanics of solids with general imperfect coherent interfaces. Arch Appl Mech 84:1409–1426CrossRef
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Esmaeili A, Javili A, Steinmann P (2016) A thermo-mechanical cohesive zone model accounting for mechanically energetic kapitza interfaces. Int J Solids Struct 92:29–44CrossRef
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Esmaeili A, Steinmann P, Javili A (2017) Coupled thermally general imperfect and mechanically coherent energetic interfaces subject to in-plane degradation. J Mech Mat Struct 12:289–312MathSciNetCrossRef
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Steinmann P, Runesson K (2021) The catalogue of computational material models: basic geometrically linear models in 1D. Springer Nature, Berlin
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Maugin GA (2000) On the universality of the thermomechanics of forces driving singular sets. Arch Appl Mech 70:31–45CrossRef
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Kienzler R, Herrmann G (2003) On the four-dimensional formalism in continuum mechanics. Acta Mech 161:103–125CrossRef
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Kienzler R, Herrmann G (2004) On conservation laws in elastodynamics. Int J Solids Struct 41:3595–3606CrossRef
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Steinmann P (2011) Geometrically nonlinear gravito-elasticity: hyperelastostatics coupled to newtonian gravitation. Int J Eng Sci 49:1452–1460MathSciNetCrossRef
Metadata
Title
Consequences of Thermodynamical 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_12

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