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On the thermomechanics of interstitial working. (English) Zbl 0582.73004
Korteweg’s constitutive assumptions, destined to describe the capillarity effects and the structure of liquid phase transition under both static and dynamic conditions, are generally, incompatible with conventional thermodynamics.
Regarding the Korteweg’s type of continuum models like special examples of elastic materials of grade \(N\), it is the aim of this paper to elaborate the theory of a thermodynamic structure comprising the non-trivial Korteweg’s type materials. The theory presented here is preserving in their standard forms the principles of linear and angular momentum balance, as well as the Clausius-Duhem inequality. Only the energy balance is modified: for each process \(\pi\) it is postulated the existence of a rate supply of mechanical energy \(u=u(X,t;n)\), called the interstitial work, defined on all \((X,t)\in B\times {\mathbb{R}}\) and all unit vectors \(n\), such that, with usual notations, the energy balance for each subdomain \(P\) of the body \(B\) has the form \[ \frac{d}{dt}\int_{P_ t}\rho (\epsilon +\frac{1}{2}\dot x\cdot \dot x)\,dv=\int_{\partial P_ t}(Tn\cdot \dot x+u-q\cdot n)\,da+\int_{P_ t}\rho (b\cdot \dot x+r)\,dv. \] An analogue of Cauchy’s theorem shows that the above balance law holds for all \(P\subset B\) iff the \(u(X,t;n)=u(X,t)\cdot n\) for any unit vector \(n\).
The local form of the equations governing this theory are deduced, and the thermodynamic consequences of this equations for the constitutive structure arising from the assumption that \(\epsilon\), \(\eta\), \(T\), \(q\), and \(u\) are smooth functions of \(F\), \(\theta\), \(\nabla F\), \(\nabla^ 2F\), \(g=\text{grad } \theta\), and \(F\) are investigated. The particular cases of: 1. General elastic materials (viscous effects are absent, i.e. \(\epsilon\), \(\eta\), \(T\), \(q\), and \(u\) are independent of \(F\)), 2. Materials of Korteweg type (i.e. \(\epsilon\), \(\eta\), \(T\), \(q\), and \(u\) are functions of density \(\rho\), d\(=\text{grad} \rho,\quad S=\text{grad}^ 2\,\rho\), \(g\), and \(L=\dot F\cdot F^{-1}\) and 3. Elastic materials of Korteweg type (i.e. materials of Korteweg type without viscous effects), are throughly discussed.
The paper is also containing a consistent introduction, pointing out the physical base of this theory and its place among the existing ones, and three welcome Appendices. The authors, well known specialists in the field, have done an excellent job writing this paper which will become a standard reference for foundations of thermodynamics. The paper is of great interest to research scientists in thermodynamics, applied thermodynamics, physicists, and engineers.
Reviewer: Gh.Gr.Ciobanu

MSC:
74A15 Thermodynamics in solid mechanics
74Axx Generalities, axiomatics, foundations of continuum mechanics of solids
80A05 Foundations of thermodynamics and heat transfer
76A05 Non-Newtonian fluids
74B10 Linear elasticity with initial stresses
74F05 Thermal effects in solid mechanics
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[1] Toupin, R. A., Elastic materials with couple-stresses. Ar · Zbl 0112.16805 · doi:10.1007/BF00253945
[2] Toupin, R. A., Theories of elasticity with couple-stress. Ar · Zbl 0131.22001 · doi:10.1007/BF00253050
[3] Gurtin, M., Thermodynamics and the possibility of spatial interaction in elastic materials. Ar · Zbl 0146.21106 · doi:10.1007/BF00253483
[4] Eringen, A. C., A unified theory of thermomechanical materi · Zbl 0139.20204 · doi:10.1016/0020-7225(66)90022-X
[5] Fixman, M., Transport coefficients in the gas critical r · Zbl 0594.05002 · doi:10.1063/1.1712302
[6] Felderhof, B. U., Dynamics of the diffuse gas-liquid interface near the · doi:10.1016/0031-8914(70)90184-9
[7] Blinowski, A., On the surface behavior of gradient-sensit · Zbl 0253.76002
[8] Blinowski, A., On the order of magnitude of the gradient-of-density dependent part of an elastic potential · Zbl 0274.76029
[9] Blinowski, A., Gradient description of capillary phenomena in multicompo · Zbl 0323.76079
[10] Blinowski, A., On the phenomenological models of capillar · Zbl 0417.73003
[11] Aifantis, E. C., & J. Serrin, The mechanical theory of fluid interfaces and Maxwell’s rule. J · doi:10.1016/0021-9797(83)90053-X
[12] Aifantis, E. C., & J. Serrin, Equilibrium solutions in the mechanical theory of fluid microstructures. J · doi:10.1016/0021-9797(83)90054-1
[13] Slemrod, M., Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Ar · Zbl 0505.76082 · doi:10.1007/BF00250857
[14] Slemrod, M., Dynamic phase transitions in a van der Waals fluid. J · Zbl 0487.76006 · doi:10.1016/0022-0396(84)90130-X
[15] Slemrod, M., An admissibility criterion for fluids exhibiting phase transitions. Nonlinear Partial Differential Equations. Ed.by J. Ball. NATO Advanced Study Institute, Plenum Press: New York, 1982.
[16] Hagan, R., & M. Slemrod, The viscosity-capillarity admissibility criterion for shocks and phase transitions. Ar · Zbl 0531.76069 · doi:10.1007/BF00963839
[17] Hagan, R., & J. Serrin, Dynamic changes of phase in a van der Waals fluid, To appear in New Perspectives in Thermodynamic, Springer-Verlag, 1985. · Zbl 0613.76074
[18] Serrin, J., The form of interfacial surfaces in Korteweg’s theory of phase equilibria. Q. Appl. Math., 41, 351–364 (1983). · Zbl 0533.76099 · doi:10.1090/qam/721427
[19] Cheverton, K. J., & M. F. Beatty, On the mathematical theory of the mechanical behavior of some non-simple materials. Ar · Zbl 0344.73010 · doi:10.1007/BF00281466
[20] Beatty, M. F., & K. J. Cheverton, The basic equations for a grade 2 material viewed as an oriente · Zbl 0347.73005
[21] Murdoch, A. I., Symmetry considerations for materials of sec · Zbl 0395.73001 · doi:10.1007/BF00040979
[22] Ericksen, J. L., Conservation laws for liquid crysta · doi:10.1122/1.548883
[23] Truesdell, C., & W. Noll, The Non-Linear Field Theories · Zbl 0779.73004
[24] Müller, I., On the entropy inequality. Ar · Zbl 0163.46701 · doi:10.1007/BF00285677
[25] Eringen, A. C., Nonlocal polar elastic conti · Zbl 0229.73006 · doi:10.1016/0020-7225(72)90070-5
[26] Eringen, A. C., & D. G. B. Edelen, On nonlocal elastic · Zbl 0247.73005 · doi:10.1016/0020-7225(72)90039-0
[27] Eringen, A. C., On nonlocal fluid mechan · Zbl 0241.76009 · doi:10.1016/0020-7225(72)90098-5
[28] Aifantis, E. C., A proposal for continuum with microstr · doi:10.1016/0093-6413(78)90047-2
[29] Coleman, B. D., & V. J. Mizel, Existence of caloric equations of state in thermody · doi:10.1063/1.1725257
[30] Noll, W., Representations of certain isotropic tenso · Zbl 0194.06401 · doi:10.1007/BF01220884
[31] Dunn, J. E., Interstitial working and a nonclassical continuum thermodynamics. To appear, New Perspectives in Thermodynamics, Springer-Verlag, 1985.
[32] Coleman, B. D., & V. J. Mizel, Thermodynamics and departures from Fourier’s law of heat conduction. Ar · Zbl 0114.44905 · doi:10.1007/BF01262695
[33] Green, A. E., & R. S. Rivlin, Simple force and stress multipoles. Arch. Rational Mech. Anal. 16, 325–353 (1964). · Zbl 0244.73005 · doi:10.1007/BF00281725
[34] Green, A. E., & R. S. Rivlin, Multipolar continuum mechanics. Arch. Rational Mech. Anal. 17, 113–147 (1964). · Zbl 0133.17604 · doi:10.1007/BF00253051
[35] Gurtin, M. E., & L. S. Vargas, On the classical theoty of reacting fluid mixtures. Ar · Zbl 0227.76011 · doi:10.1007/BF00251451
[36] Olver, P. J., Conservation laws and null divergences. Math · Zbl 0556.35021 · doi:10.1017/S030500410000092X
[37] Olver, P. J., Conservation laws and null divergences. II Nonnegative divergences. Math. Proc. Camb. Phil. Soc. To appear. · Zbl 0561.35011
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