Continuum body data, the fixed inertial frame and the dimensions of the fields

(a) Framework and frame. The framework is the non-relativistic classical mechanics of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure with the inertial frames, stationary observers and allowed changes of frame of Inertial frames, stationary observers and the adopted Galilean transformations between frames; the particle worldline conventions of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses are used for the labeled points of a body, and dimensions and reference units (metre, second, kilogram) are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Throughout this page one inertial frame $F$ with adapted coordinates $(t,x)$ of the affine time space Affine time space with Euclidean simultaneous slices and adapted charts is fixed, and each simultaneous spatial slice is identified with ${\mathbb{R}}^3$ carrying its standard Euclidean structure. The time coordinate $t$ is the model's time in seconds, $[t]=\tau$, and the spatial components $x_i$ are positions in metres, $[x_i]=\mathrm{\ell }$. All fields below are read in this one frame.

(b) Body data. A continuum body data set in $F$ is a triple $(B,O,I)$ consisting of

1. a material region $B\subseteq {\mathbb{R}}^3$: a nonempty, bounded, open set (Material regions, densities and their mass);
2. an open set $O\subseteq {\mathbb{R}}^3$ with $\bar{B}\subseteq O$, the reference chart domain;
3. a nonempty open interval $I\subseteq \mathbb{R}$, the time window (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

The points $X\in B$ are the material labels of the body and $B$ is its reference placement. The reference placement is bookkeeping for the labels and not an observable: it is used only to name the points of the body, and the full-motion branch concerns spatial fields on the current regions; the linear elastic branch instead uses reference fields. The reference chart's Euclidean structure fixes the derivatives of these reference fields. It does not assign an intrinsic mass or force to a label.

(c) Field slots and dimensions. These slots are conditional data, not assertions of physical existence. Whenever a motion is specified, use exactly Continuum motions, spatial velocity, material acceleration and swept regions: a $C^2$ map $\chi :I\times O\to {\mathbb{R}}^3$, with each ${\chi }_t$ injective and $D_X{\chi }_t$ invertible, and $\det D_X{\chi }_t>0$ on $I\times B$. Put ${\mathrm{\Omega }}_t={\chi }_t(B)$ and $\mathcal{U}=\{(t,x):t\in I,x\in {\chi }_t(O)\}$, the open spacetime domain of that supplier. Spatial fields below are defined on $\mathcal{U}$ or on a stated open subdomain containing the current events of interest; additional regularity is specified wherever needed. The dimensions use Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

1. Density is a scalar map $\rho :\mathcal{U}\to (0,\mathrm{\infty })$ of class $C^1$, read as mass per unit spatial volume, $[\rho ]=\mu {\mathrm{\ell }}^{-3}$.
2. Spatial velocity is the $C^1$ map $u:\mathcal{U}\to {\mathbb{R}}^3$ defined by $u(t,\chi (t,X))={\mathrm{\partial }}_t\chi (t,X)$, $[u]=\mathrm{\ell }{\tau }^{-1}$. For any $C^1$ scalar or vector field $f$ on this domain define $D_{t}f={\mathrm{\partial }}_{t}f+D_{x}fu$, componentwise for vectors, as in Euclidean spatial calculus: gradient, divergence and material derivative. Material acceleration is the continuous map $a:=D_{t}u:\mathcal{U}\to {\mathbb{R}}^3$, $[a]=\mathrm{\ell }{\tau }^{-2}$.
3. Body force per unit mass is a continuous map $b:\mathcal{U}\to {\mathbb{R}}^3$, $[b]=\mathrm{\ell }{\tau }^{-2}$; its force per unit volume is $\rho b$, of dimension $\mu {\mathrm{\ell }}^{-2}{\tau }^{-2}$.
4. With $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\parallel }n{\mathrm{\parallel }}_2=1\}$, traction is a map $t:\mathcal{U}\times S^2\to {\mathbb{R}}^3$, read as force per unit area acting on the side opposite to the direction $n$. A stress matrix is a matrix field $\sigma :\mathcal{U}\to M_3(\mathbb{R})$ representing it by $t(s,x,n)=\sigma (s,x)n$ when such a field exists; existence and regularity are separate hypotheses. Here $[t]=[\sigma ]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$ (SI $\mathrm{N}{\mathrm{m}}^{-2}$).
5. Displacement relative to the stationary reference placement is $\xi :I\times O\to {\mathbb{R}}^3$, $\xi (t,X)=\chi (t,X)-X$, $[\xi ]=\mathrm{\ell }$; its linear strain is the symmetric matrix field $e=(D_X\xi +(D_X\xi {)}^{\mathsf{T}})\mathrm{/}2$, of dimension $1$. A linearized branch may instead prescribe a displacement field with these same domain and strain conventions; identifying it with an exact motion requires the motion hypotheses above.
6. Elastic moduli $\lambda ,{\mu }_s$ are real scalar model parameters; pressure is a scalar field $p:\mathcal{U}\to \mathbb{R}$ of class $C^1$, read as the coefficient in the isotropic stress $-p{I}_3$. Their dimension is $\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$. Dynamic viscosity $\eta$ is a real scalar model parameter of dimension $\mu {\mathrm{\ell }}^{-1}{\tau }^{-1}$ (SI $\mathrm{P}\mathrm{a}\mathrm{s}$). Signs and constitutive uses of these parameters are specified by each branch.
7. Wave speed $c>0$ is a scalar parameter of a specified wave model, $[c]=\mathrm{\ell }{\tau }^{-1}$; its construction from that model's parameters is stated in the relevant branch, rather than inferred from dimensions.

(d) Frame and reference-label conventions. Write a Galilean change as $G(t,x)=(t^{\mathrm{\prime }},x^{\mathrm{\prime }})=(t+\beta ,Rx+vt+a_0),R\in SO(3),$ with the conventions of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes. At corresponding events, a scalar field $q$ (for example, density $\rho$ or pressure $p$) transforms by $q^{\mathrm{\prime }}\circ G=q$. Spatial vectors such as acceleration and force per unit mass transform by $a^{\mathrm{\prime }}\circ G=Ra$, $b^{\mathrm{\prime }}\circ G=Rb$, while velocity has the boost term $u^{\mathrm{\prime }}\circ G=Ru+v$. Normals transform by $n^{\mathrm{\prime }}=Rn$, traction by $t^{\mathrm{\prime }}(G(t,x),Rn)=Rt(t,x,n),$ and stress by ${\sigma }^{\mathrm{\prime }}\circ G=R\sigma R^{\mathsf{T}}$. Rotations preserve spatial volume and area. Spatial differentiation is at fixed frame time; these rules give $D_{x^{\mathrm{\prime }}}{u}^{\mathrm{\prime }}=R(D_{x}u)R^{\mathsf{T}}$, ${\mathrm{div}}_{x^{\mathrm{\prime }}}{u}^{\mathrm{\prime }}={\mathrm{div}}_{x}u$ and $D_{t^{\mathrm{\prime }}}{u}^{\mathrm{\prime }}=R{D}_{t}u$ at corresponding events. In particular, ${\sigma }^{\mathrm{\prime }}{n}^{\mathrm{\prime }}=R\sigma n$, so traction representation is preserved.

Here the local mass and momentum equations mean precisely $D_t\rho +\rho {\mathrm{div}}_{x}u=0,\rho D_{t}u=\rho b+g,g_i:={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{\sigma }_{ij},$ with $\rho ,u,\sigma$ of class $C^1$ on a common open subdomain of $\mathcal{U}$ and $b$ continuous there, as in Local mass balance: conservation on every material subregion implies ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$ and Local momentum balance: $\rho D_{t}u=\rho b+\mathrm{div}\sigma$ from the integral balance on admissible material subregions. These are equation templates, not adopted laws in this definition. The chain rule applied to $G^{-1}$ gives $(D_{t^{\mathrm{\prime }}}{\rho }^{\mathrm{\prime }})\circ G=D_t\rho$ and $g^{\mathrm{\prime }}\circ G=Rg$ (in the latter identity the sum over spatial indices uses $R^{\mathsf{T}}R=I_3$). Thus the mass residual is unchanged and the momentum residual is multiplied by $R$; each equation holds exactly when its transformed equation holds.

For the integral forms, use the exact setting of those two suppliers, including ${\mathrm{A}\mathrm{C}}_{\omega }$: $J_0$ has compact closure in $I$, the common open field domain contains its swept closure, and $B_0\subseteq B$ is nonempty bounded open with $\overline{B_0}\subseteq B$. Write $V_t={\chi }_t(B_0)$ for the moving region, and put $M(t):={\int }_{V_t}\rho dx,P(t):={\int }_{V_t}\rho udx,\mathcal{F}(t):={\int }_{V_t}\rho bdx+{\int }_{\mathrm{\partial }{V}_t}\sigma \nu dS\mathrm{.}$ The volume integrals are Lebesgue integrals, componentwise for vectors; the force expression is used only when $V_t$ is a bounded $C^1$ domain, with outward unit normal $\nu$ and the surface integral of Local momentum balance: $\rho D_{t}u=\rho b+\mathrm{div}\sigma$ from the integral balance on admissible material subregions. The equations are $\dot{M}=0$ and $\dot{P}=\mathcal{F}$, on the respective admissible regions and times of the suppliers. At $t^{\mathrm{\prime }}=t+\beta$ the corresponding region is $R{V}_t+vt+a_0$. Volume and surface measure preservation and ${\nu }^{\mathrm{\prime }}=R\nu$ give $M^{\mathrm{\prime }}(t^{\mathrm{\prime }})=M(t)$, $P^{\mathrm{\prime }}(t^{\mathrm{\prime }})=RP(t)+vM(t)$ and ${\mathcal{F}}^{\mathrm{\prime }}(t^{\mathrm{\prime }})=R\mathcal{F}(t)$, where primes here label the transformed quantities. Consequently $d{P}^{\mathrm{\prime }}\mathrm{/}d{t}^{\mathrm{\prime }}=R\dot{P}+v\dot{M}$: the integral momentum equation retains its form together with mass conservation. No covariance of momentum alone without the mass premise is asserted.

For a reference-label chart choose a fixed reference time $t_{\mathrm{r}}\in I$ and set $X^{\mathrm{\prime }}=RX+v{t}_{\mathrm{r}}+a_0$. This is a time-independent relabeling, with $B^{\mathrm{\prime }},O^{\mathrm{\prime }}$ the images of $B,O$, and ${\chi }^{\mathrm{\prime }}(t^{\mathrm{\prime }},X^{\mathrm{\prime }})=R\chi (t,X)+vt+a_0,t=t^{\mathrm{\prime }}-\beta \mathrm{.}$ A material scalar transforms by composition with the inverse label/time map; a material vector by that composition followed by $R$. In the linear elastic branch the stress-free comparison placement $X$ in $F$ transforms to the uniformly translating comparison placement $RX+vt+a_0$ in $F^{\mathrm{\prime }}$. Define displacement as the difference from this same comparison placement, so ${\xi }^{\mathrm{\prime }}(t^{\mathrm{\prime }},X^{\mathrm{\prime }})=R\xi (t,X),D_{X^{\mathrm{\prime }}}{\xi }^{\mathrm{\prime }}=R(D_X\xi )R^{\mathsf{T}},e^{\mathrm{\prime }}=Re{R}^{\mathsf{T}}\mathrm{.}$ Time derivatives here are at fixed labels. Consequently ${\mathrm{\partial }}_{t^{\mathrm{\prime }}}^2{\xi }^{\mathrm{\prime }}=R{\mathrm{\partial }}_t^2\xi$, while the label's coordinate velocity in the boosted description is $v+R{\mathrm{\partial }}_t\xi$. In the stationary comparison frame its coordinate velocity is ${\mathrm{\partial }}_t\xi$; for a finite motion the spatial velocity is related by $u(t,\chi (t,X))={\mathrm{\partial }}_t\xi (t,X)$. Constant isotropic elastic moduli and viscosity are scalars. Precisely, the isotropic strain-stress law of The small-strain isotropic linear elastic constitutive law and its strain energy, applied at each time to a $C^2$ reference displacement, is ${\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}=\lambda (\mathrm{tr}e)I_3+2{\mu }_{s}e$ with constant ${\mu }_s>0$ and $\lambda +2{\mu }_s\mathrm{/}3>0$. Since $\mathrm{tr}(Re{R}^{\mathsf{T}})=\mathrm{tr}e$ and $R{I}_3{R}^{\mathsf{T}}=I_3$, its rotated stress is $R{\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}{R}^{\mathsf{T}}=\lambda (\mathrm{tr}{e}^{\mathrm{\prime }})I_3+2{\mu }_s{e}^{\mathrm{\prime }}$. Thus it retains its form. These are conditional coordinate-reading rules, not a finite-strain approximation theorem or a new constitutive assumption.

• No dynamical content. Nothing above asserts that any body moves, that any mass, force or stress exists, or that any field of the list occurs in a model. The list fixes names, slots and dimensions for the postulates and theorems that follow, and the dimension computations are consistency conventions in the sense of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, not physical laws.

• Reference versus current placement. The choice of $B$ and $O$ is bookkeeping for the labels. The full-motion and reference-field branches use their respective domains above. Clause (d) specifies the affine relabeling convention and the conditional transformation calculations; it does not identify a reference strain with a finite-strain observable.

• Model quantities versus measured quantities. The fields introduced here are model quantities read in a frame. A measured mass, density or force is related to them only through an explicit measurement model and is not asserted by this definition; no empirical value is recorded on this page.