Continua and Generalities About Their Equations

Abstract

A homogeneous continuum, chemically inert, in d dimensions is described by:

• (a) a region Ω in ambient space (Ω ⊂ ℝ^d), which is the occupied volume;

• (b) a function P → ρ(P) > 0, defined on Ω, giving the mass density;

• (c) a function P → T(P) defining the temperature;

• (d) a function P → s(P) defining the entropy density (per unit mass);

• (e) a function P → δ(P) defining the displacement with respect to a reference configuration;

• (f) a function P → u(P) defining the velocity field;

• (g) an equation of state relating T(P), s(P), ρ(P);

• (h) a stress tensor $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau }$, also denoted (τ _ij ), giving the force per unit surface that the part of the continuum in contact with an ideal surface element dσ, with normal vector n, on the side of n, exercises on the part of the continuum in contact with dσ on the side opposite to n, via the formula

  $$d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} d\sigma \;{(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} )_i} = \sum\nolimits_{j = 1}^d {{\tau _{ij}}} {n_j}$$

  (1.1.1)

• (i) a thermal conductivity tensor $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}$, giving the quantity of heat traversing the surface element dσ; in the direction of n per unit time via the formula

  $$dQ = - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial } T\;d\sigma $$

  (1.1.2)

• (l) a volume force density P → g(P);

• (m) a relation expressing the stress and conductivity tensors as functions of the observables δ, u, ρ, T, s.