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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.

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 |

### Keywords:

Korteweg’s constitutive assumptions; capillarity effects; structure of liquid phase transition; static and dynamic conditions; incompatible with conventional thermodynamics; principles of linear and angular momentum balance; Clausius-Duhem inequality; energy balance is modified; interstitial work; General elastic materials; Materials of Korteweg type
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\textit{J. E. Dunn} and \textit{J. Serrin}, Arch. Ration. Mech. Anal. 88, 95--133 (1985; Zbl 0582.73004)

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