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Thermodynamics of Binary Solid–Fluid Cosserat Mixtures

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Abstract

It is demonstrated in this chapter, how complex it is to deduce a saturated binary solid–fluid Cosserat mixture model that is in conformity with the second law of thermodynamics and sufficiently detailed to be ready for application in fluid dynamics. The second law is formulated for open systems using the Clausius –Duhem inequality without mass and energy production under phase change for class II mixtures of elastic solids and viscoelastic fluids. It turns out that even with all these restrictions the detailed exploitation of the entropy inequality is a rather involved endeavor. Inferences pertain to extensive functional restrictions of the fluid- and solid- free energies and allow determination of the constitutive quantities in terms of the latter in thermodynamic equilibrium and small deviations from it. The theory is presented for four models of compressible–incompressible fluid–solid constituents. Finally, explicit representations are given for the free energies and for the constitutive quantities that are obtained from them via differentiation processes

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Notes

  1. 1.

    For a brief biographical sketch of Bernard D. Coleman (1930–2018), see Fig.  24.1 .

  2. 2.

    For a brief biographical sketch of Walter Noll (1925–2017), see Fig.  24.2 .

  3. 3.

    For brief biographies of

    • Rudolf Julius Emanuel Clausius , see Fig. 17.8, Vol. 2 of this treatise, p. 330 [42];

    • Pierre Maurice Marie Duhem , see Fig. 17.16, Vol. 2 of this treatise, p. 343 [42];

    • Walter Noll, see [43];

    • Clifford Ambrose Truesdell , see Fig. 22.1, this volume, p. 40.

  4. 4.

    A separate thermodynamic formulation on granular-fluid continua will be given in Chap. 30.

  5. 5.

    Hydrophilic polymers swell under water (vapor) adsorption.

  6. 6.

    In the formulae (24.120)–(24.123), a subscript \((\cdot )_1\) has been introduced to identify that the so subscripted quantities are based on the rule of phase separation rather than the rule of aequipresence.

  7. 7.

    In the most general case of (24.129) the g-functions are fourth-order tensors over \({\mathbb R}^{3}\). Here, by assuming the simplest representation of isotropy, they are merely scalars.

  8. 8.

    See e.g., [40], Chap. 1, homework 8, p. 43.

  9. 9.

    Pertinent early memoirs are as well by Eringen [28, 29], Grioli [38, 39], Mindlin and Tiersten [50], Neuber [54], Schäfer [63,64,65], Smith [69, 70], Soos [71, 72], Toupin [76], Wozniak [82], Wyrwinski [83], and many others, see e.g., [58].

  10. 10.

    A detailed explanation of these arguments is given in Vol. 2 Chap. 18 ‘Thermodynamics – Field Formulation’ of this treatise [42].

  11. 11.

    This means, constitutive equations are invariant with respect to arbitrary rotations.

  12. 12.

    Alternatively, (24.178) and (24.179) can also be formulated for the right Cauchy –Green deformation tensor \({\varvec{C}}={\varvec{F}}^{T}{\varvec{F}}\) ; the eigenvalues \(\lambda _{i}\) \((i=1,2,3)\) are the same for \(\sqrt{\varvec{C}}\) as for \(\sqrt{{\varvec{B}}}\), see e.g., [40].

  13. 13.

    The representation (24.186), (24.187) as quadratic polynomial tensor relations of \({\varvec{I}}, {\varvec{B}}, {\varvec{B}}{\varvec{B}}\) is a consequence of the Cayley –Hamilton theorem for any smooth function.

  14. 14.

    To prove this, take the trace of the term \(\{\cdot \}\) in the first curly brackets of (24.188). This yields

    $$\begin{aligned} I\!I_{\hat{{\varvec{B}}}} \frac{\partial \,\hat{W}}{\partial \,I\!I_{\hat{{\varvec{B}}}}} - I_{\hat{{\varvec{B}}}}\frac{\partial \,\hat{W}}{\partial \,I_{\hat{{\varvec{B}}}}} + \frac{1}{I\!I\!I_{\hat{\varvec{B}}}^{1/3}}\frac{\partial \,\hat{W}}{\partial \,I_{\hat{\varvec{B}}}}I_{{\varvec{B}}} -I\!I\!I_{\hat{\varvec{B}}}^{1/3} \frac{\partial \,\hat{W}}{\partial \,I\!I_{\hat{\varvec{B}}}}I_{{\varvec{B}}^{-1}} {\mathop {=}\limits ^{\mathrm {def.}}} {\mathfrak Q}. \end{aligned}$$

    Owing to (24.181)\(_{1}\), the second and third term of this expression cancel each other. To conclude the same for the remaining terms, \(I_{{\varvec{B}}^{-1}}\) must be expressed in terms of the invariants of \({\varvec{B}}\). This follows from the Cayley –Hamilton theorem

    $$\begin{aligned} {\varvec{B}}^{3} - I_{{\varvec{B}}}{\varvec{B}}^{2} + I\!I_{{\varvec{B}}}{\varvec{B}} - I\!I\!I_{\varvec{B}}{\varvec{I}} \equiv 0, \end{aligned}$$

    from which one easily deduces

    $$\begin{aligned}&{\varvec{B}}^{2} - I_{{\varvec{B}}}{\varvec{B}}+ I\!I_{\varvec{B}}{\varvec{I}} - I\!I\!I_{\varvec{B}}{\varvec{B}}^{-1} = 0 ,\\&\Longrightarrow \quad&{\varvec{B}}^{-1} = \frac{1}{I\!I\!I_{{\varvec{B}}}}{\varvec{B}}^{2} - \frac{I_{{\varvec{B}}}}{I\!I\!I_{{\varvec{B}}}}{\varvec{B}} + \frac{I\!I_{{\varvec{B}}}}{I\!I\!I_{{\varvec{B}}}}{\varvec{I}}. \end{aligned}$$

    Forming the trace of this equation yields

    $$\begin{aligned} \mathrm {tr}{\varvec{B}}^{-1} \equiv I_{{\varvec{B}}^{-1}} = \frac{1}{I\!I\!I_{{\varvec{B}}}}\left\{ \underbrace{ I_{{\varvec{B}}^{2}} - (I_{\varvec{B}})^{2}}_{ - 2I\!I_{{\varvec{B}}} } +3I\!I_{{\varvec{B}}} \right\} = \frac{I\!I_{{\varvec{B}}}}{I\!I\!I_{{\varvec{B}}}} {\mathop {=}\limits ^{(24.181)_{2}}} I\!I_{\hat{{\varvec{B}}}}I\!I\!I_{{\varvec{B}}}^{-1/3}, \end{aligned}$$

    which proves now that \({\mathfrak Q} = 0\).

  15. 15.

    In [16] also a condition on \(\hat{W}\) is stated from which the shear modulus follows, but that condition does not affect U.

  16. 16.

    This latter condition is not satisfied by the free energy function (24.199).

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  1. c/o Laboratory of Hydraulics, Hydrology, Glaciology, ETH Zürich, Zürich, Switzerland

    Kolumban Hutter

  2. Department of Mechanical Engineering, Darmstadt University of Technology, Darmstadt, Hessen, Germany

    Yongqi Wang

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Hutter, K., Wang, Y. (2018). Thermodynamics of Binary Solid–Fluid Cosserat Mixtures. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77745-0_24

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