Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance

Adopt as physical assumptions of the continuum branch the following statements, in the framework, frame and body data of Continuum body data, the fixed inertial frame and the dimensions of the fields.

(i) Model data. A model of the branch is carried by a continuum body data set $(B,O,I)$ of Continuum body data, the fixed inertial frame and the dimensions of the fields in the fixed inertial frame $F$: a material region $B$, a reference chart domain $O$ with $\bar{B}\subseteq O$, and a time window $I$. The labels $X\in B$ are the material points of the body.

(ii) Motion. A motion of the model is a map

$\chi :I\times O⟶{\mathbb{R}}^3$

which is a motion of the material region $B$ on $I$ in the sense of Continuum motions, spatial velocity, material acceleration and swept regions: $\chi$ is of class $C^2$; for every $t\in I$ the map ${\chi }_t:O\to {\mathbb{R}}^3$ is of class $C^1$, injective and has invertible total derivative at every point of $O$; and the Jacobian $J(t,X)=\det (D_X{\chi }_t(X))$ satisfies $J>0$ on $I\times B$. The current region is ${\mathrm{\Omega }}_t:={\chi }_t(B)$, and the model's spatial velocity field $u$ and material acceleration $a=D_{t}u$ are those of Continuum motions, spatial velocity, material acceleration and swept regions and Euclidean spatial calculus: gradient, divergence and material derivative.

(iii) Worldlines of labels. For every label $X\in B$ the curve ${\gamma }_X:I\to A$, ${\gamma }_X(t):=(t,\chi (t,X))$, is a worldline of the model in the sense of clauses (i)-(iv) of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses: it is of class $C^2$ in the adapted coordinates of $F$, its parameter is the frame time ($\tau ({\gamma }_X(t)-o)=t$), its description in another inertial frame is the reparametrized curve induced by the frame change, and its coordinate velocity and acceleration are read by Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors. The motion is thus a family of worldlines, one for each label, and the spatial velocity field satisfies $u(t,\chi (t,X))={\mathrm{\partial }}_t\chi (t,X)$; no mass is attached to a label, mass entering only through a density field.

(iv) Continuum idealization. The body is described as a continuum: every label is a point and the fields are defined at every point of the current regions. The following are excluded by this postulate, and no statement of this page covers them: fracture, cutting or interpenetration (injectivity and $J>0$); shocks, phase boundaries and material interfaces inside the motion (the motion is $C^2$ with $J>0$ everywhere on $I\times B$); diffusion or exchange of material between labels; chemical reactions, phase change and mass sources; electromagnetic, thermal and radiative couplings; and microscopic or molecular structure. The smoothness classes of the fields are stated in each later item; wherever a weaker regularity is required it is stated there, and the classical transport and balance identities of this page are used only under their own hypotheses.

(v) Frame covariance. A model is frame-covariant when, for every change of inertial frame $F\to F^{\mathrm{\prime }}$ given by the Galilean transformation of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes, the description of the same motion in $F^{\mathrm{\prime }}$ is the Galilean transform of the description in $F$, and the model's balance and constitutive laws have the same form when written in $F^{\mathrm{\prime }}$ with the correspondingly transformed fields (velocities transformed by the boost, accelerations rotated as there stated). The branch requires frame covariance of every model; for each law adopted below, the covariance check is carried out or the law is stated only in the fixed frame.

(vi) Status. These are adopted assumptions of the model. They are not derived from mechanics and they assert nothing about any particular material. The continuum idealization is used in practice when the fields vary on spatial scales large compared with the microscopic spacing of the material and the material admits a representative volume element; those microscopic conditions are not formalized here and no quantitative threshold is asserted. Empirical adequacy of the idealization for a given material is a matter of comparison with observations and is not established by this postulate.