Abstract
We first review the principles of Classical Thermodynamics (see also Appendix D), and proceed to give an alternative formulation of Thermodynamics in the context of a true dynamical process.
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Notes
- 1.
The rate of the internal mechanical energy \(d{\mathcal{U}}_{\sigma }/dt\)due to stress is frequently referred to as the rate of internal work and is written as \(d{\mathcal{W}}_{\sigma }/dt\). The rate of its specific value \(d{u}_{\sigma }(\mathbf{x},t)/dt\)is similarly written as \(d{w}_{\sigma }(\mathbf{x},t)/dt\).
- 2.
Since the stretch tensor \(\mathbf{D}\)may be decomposed into a reversible (i.e., elastic) part \({\mathbf{D}}^{e}\)and an irreversible (i.e., inelastic) part \({\mathbf{D}}^{i}\)(\(\mathbf{D}\! =\!{ \mathbf{D}}^{e}\! +\!{ \mathbf{D}}^{i}\)), (3.26) is valid. However this decomposition is not unique. We do not discuss the details here (see Raniecki and Nguyen 2005).
- 3.
\(\sigma \! =\! {\sigma }_{\mathit{ij}}\,{\mathbf{e\!}}_{i} \otimes {\mathbf{e\!}}_{j},\ \dot{\mathbf{E}}\! =\!\dot{ {E}}_{\mathit{IJ}}\,{\mathbf{E\!}}_{I} \otimes {\mathbf{E\!}}_{J}\). From (??) we obtain
$$\qquad {\mathbf{F}}^{-T}\dot{\mathbf{E}}{\mathbf{F}}^{-1} = \frac{\partial {X}_{I}} {\partial {x}_{i}} \dot{{E}}_{IJ}\frac{\partial {X}_{J}} {\partial {x}_{j}}{ \mathbf{e\!}}_{i} \otimes {\mathbf{e\!}}_{j},{\mathbf{F}}^{-1}\sigma {\mathbf{F}}^{-T} = \frac{\partial {X}_{I}} {\partial {x}_{i}} {\sigma }_{ij}\frac{\partial {X}_{J}} {\partial {x}_{j}}{ \mathbf{E\!}}_{I} \otimes {\mathbf{E\!}}_{J}$$The inner product of second order tensors \(\mathbf{A},\ \mathbf{B}\)is given by \(\mathbf{A}\! :\! \mathbf{B}\! =\! \text{ t}r({\mathbf{A}}^{T}\mathbf{B})\); thus (3.34) can be proved.
- 4.
Suppose that we have a function fsuch that s † = f(s ∗ ), and set
$${T}^{\dag } = \frac{{T}^{{_\ast}}} {df({s}^{{_\ast}})/d{s}^{{_\ast}}}.$$This is equivalent to
$${T}^{\dag }\,d{s}^{\dag } = {T}^{{_\ast}}\,d{s}^{{_\ast}}.$$Thus the representation (3.57) is not unique.
- 5.
If the domain of a function is (a subset of) one dimensional real number space ℝ, it is said to be a ‘functional’. The potential functions, such as the internal energy, are a typical functional.
- 6.
In thermodynamics \({\frac{\partial A} {\partial x} \biggr |}_{a,b}\)implies differentiation of Awith respect to xunder a, bconstant.
- 7.
The differential appeared in (3.175)
$$\qquad { \frac{\partial u} {\partial {c}_{\alpha }}\biggr |}_{{\epsilon }^{e},\,s}$$and is sometimes referred to as the partial mass value (this case, partial mass internal energy) (cf. Slattery 1999, p. 447).
- 8.
In some textbooks (3.198) is rewritten by using the changes of enthalpy Δhand the Gibbs free energy Δgbefore and after the reaction, and the Gibbs-Helmholtz relation is given by
$$\qquad \frac{\partial \,} {\partial T}\left (\frac{\Delta g} {T} \right ) = -\frac{\Delta h} {{T}^{2}}$$Time-differentiation of (3.198) suggests that the above form is not accurate.
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© 2012 Springer-Verlag Berlin Heidelberg
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Ichikawa, Y., Selvadurai, A.P.S. (2012). Non-equilibrium Thermodynamics. In: Transport Phenomena in Porous Media. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25333-1_3
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DOI: https://doi.org/10.1007/978-3-642-25333-1_3
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