The density field of the continuum model and the mass of a material subregion

Fix body data $(B,O,I)$ and a motion $\chi$ of the model on $I$ in the sense of Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance; write ${\mathrm{\Omega }}_t={\chi }_t(B)$ for the current regions and, for an open interval $J_0$ with $\overline{J_0}\subseteq I$, let $\mathrm{\Sigma }=\{(t,x):t\in \overline{J_0},x\in {\chi }_t(\bar{B})\}$ be the swept region (Continuum motions, spatial velocity, material acceleration and swept regions).

A density field of the model is a function

$\rho :U⟶(0,\mathrm{\infty })$

on an open set $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ containing the full swept set $\{(t,x):t\in I,x\in {\chi }_t(\bar{B})\}$ (and hence $\mathrm{\Sigma }$ for every compactly contained $J_0$), whose physical reading is mass per unit volume: its dimension is $[\rho ]=\mu {\mathrm{\ell }}^{-3}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, Continuum body data, the fixed inertial frame and the dimensions of the fields). The standing regularity of the balance items of this page is that $\rho$ is of class $C^1$ on $U$ ($C^k$ Euclidean maps and diffeomorphisms); where a weaker regularity is used it is stated in the item that uses it. Strict positivity and $C^1$ regularity are part of the field class of this branch; the mathematical material-region notion that underlies it admits any nonnegative integrable density (Material regions, densities and their mass) and is used here only to define the mass integral.

A material subregion at the level of labels is a nonempty bounded open $B^{\mathrm{\prime }}\subseteq B$ with $\overline{B^{\mathrm{\prime }}}\subseteq B$ (Material regions, densities and their mass); the same $\chi$ is a motion of $B^{\mathrm{\prime }}$. For the Lebesgue mass construction assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the density/mass part of Material regions, densities and their mass; the positive $C^1$ field slot and the open-region predicate themselves require no choice axiom. For $t\in I$ the model mass of $B^{\mathrm{\prime }}$ at time $t$ is the Lebesgue integral

$m(t,B^{\mathrm{\prime }}):={\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx$

(Integral over a measurable subset). Its dimension is $\mu$ and it is a number computed from the model fields; whether it equals a measured mass is not asserted here, and a comparison with an observed mass requires an explicit measurement model, calibration and uncertainty statements that are not part of this definition.

• No conservation is asserted. The definition attaches a number $m(t,B^{\mathrm{\prime }})$ to the model at each time. That this number is constant in $t$ for every material subregion is the separate mass-balance assumption of this page, adopted as a postulate and not a consequence of the definitions.

• Measured density is not the model field. A measured density is the result of a procedure (massing a sample of finite size, imaging, or inferring from scattering) with its own resolution, calibration and uncertainty. A model density field is defined at every point of the current region; identifying the two requires a measurement model and sampling assumptions, and the identification is only as good as those assumptions. No measured value is recorded by this definition.

• The velocity field used in the mass integral is the motion's. The mass integral and all later rates of change refer to the current regions ${\mathrm{\Omega }}_t$ of the one motion $\chi$; the definition does not introduce a second, independently chosen region or velocity.