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Chapter VIII – Nonlinear Viscoelasticity
[When the conservation equations for mass, chemical species, momentum and energy were derived in the previous chapter, it became soon evident that the number of unknown functions was far larger than that of the equations. To allow the closure of the problem some quantities need to be related to others and to the properties of matter, and these are the diffusive mass fluxes, jp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {j}^{\left( p\right) }$$\end{document}, the deviatoric stress tensor, τjk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{jk}$$\end{document}, the internal energy per unity of mass, u^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{u}$$\end{document} (or the specific enthalpy, h^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{h}$$\end{document}) and the heat flux, q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {q}$$\end{document}. The laws that describe these quantities are known as constitutive equations, and in thermo-fluids they are inherently empirical, although they must satisfy some requirement based upon first principles, like the condition of material objectivity (material properties must be independent of observer), the symmetry properties of a material body and the law of thermodynamics (particularly, the entropy inequality).]
Published: Jul 1, 2020
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