Classical continua with explicit constitutive assumptions

(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.

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.

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.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the general Lebesgue mass integrals of The density field of the continuum model and the mass of a material subregion; this qualifies the mathematical mass construction, not the choice-free kinematics.

Adopt the following assumption for every model of the continuum kinematic branch of Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance with density field of The density field of the continuum model and the mass of a material subregion.

(i) Mass balance. Let $\chi$ be a motion of the model on the time window $I$, let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$, and let $B^{\mathrm{\prime }}\subseteq B$ be a material subregion (nonempty bounded open with $\overline{B^{\mathrm{\prime }}}\subseteq B$). Then the model mass of $B^{\mathrm{\prime }}$,

$m(t,B^{\mathrm{\prime }})={\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx(t\in J_0),$

is constant in $t$:

$$\begin{array}{cccc}& \frac{d}{dt}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx=0(t\in J_0)\mathrm{.}& & \text{(CM)}\end{array}$$

The assumption is imposed for every material subregion and every time in the window; the family of subregions is not restricted to a selected finite list.

(ii) Reading and exclusions. The assumption says that a label carries its matter permanently: the material of the body is a closed mechanical system during the motion. It is not adopted, and no statement of this page covers: open systems in which material is added to or removed from the body; diffusion, mixing or relative transport of distinct constituents between labels; chemical reactions, phase change or mass exchange between constituents; mass sources or sinks distributed in the body; and relativistic mass-energy conversion. In such situations the balance above can fail and a different balance with source terms is required.

(iii) Empirical status. The assumption is the classical conservation of mass, adopted here as model structure. Comparisons of predictions with observed masses of prepared closed systems can support it under stated apparatus and calibration assumptions; such comparisons are evidence about the combined model and apparatus, not a deductive proof of the principle, and no measurement is asserted or used by this postulate.

(iv) Status of the consequences. The local continuity equation derived on this page is a conditional theorem: it follows from (CM) together with the declared regularity of the fields and the transport identity of the kinematic branch. If (CM) is replaced by a balance with sources, the local equation acquires the corresponding source terms and none of the present deductions applies unchanged.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as used by the transport theorem cited below. Let $(B,O,I)$ be body data of Continuum body data, the fixed inertial frame and the dimensions of the fields, let $\chi$ be a motion of the model on $I$ with spatial velocity $u$ and material acceleration $a=D_{t}u$ (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum motions, spatial velocity, material acceleration and swept regions). Let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$, put $\mathrm{\Sigma }:=\{(t,x):t\in \overline{J_0},x\in {\chi }_t(\bar{B})\}$, and let $f:U\to \mathbb{R}$ be of class $C^1$ on an open set $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ containing $\mathrm{\Sigma }$ ($C^k$ Euclidean maps and diffeomorphisms). Then:

(i) Transport identity. The function $F(t):={\int }_{{\chi }_t(B)}f(t,x)dx$ is defined and differentiable on $J_0$, and for every $t\in J_0$

$F^{\mathrm{\prime }}(t)={\int }_{{\chi }_t(B)}({\mathrm{\partial }}_{t}f+\mathrm{div}(fu))(t,x)dx\mathrm{.}$

The identity holds componentwise for a $C^1$ vector field $f$.

(ii) Material derivative along worldlines. If $f$ is the spatial description of a material field $F_{\mathrm{m}}(t,X):=f(t,\chi (t,X))$, then for every $X\in B$ the function $t\mapsto F_{\mathrm{m}}(t,X)$ is differentiable on $J_0$ and

$\frac{d}{dt}{F}_{\mathrm{m}}(t,X)=D_{t}f(t,\chi (t,X))=({\mathrm{\partial }}_{t}f+\mathrm{\nabla }f\cdot u)(t,\chi (t,X)),$

with $D_t$ and the gradient of Euclidean spatial calculus: gradient, divergence and material derivative. Writing $\mathrm{div}(fu)=\mathrm{\nabla }f\cdot u+f\mathrm{div}u$, the integrand of (i) is $D_{t}f+f\mathrm{div}u$ at the corresponding current point.

The statement is conditional on the kinematic model and on the compact containment $\overline{J_0}\subseteq I$; it asserts no conservation law, no equation of motion and no empirical value.

• Hypotheses are used exactly. Positive Jacobian and $C^2$ regularity of the motion are used to instantiate the transport theorem and the chain rule; at each time in $J_0$, a compact inner time cylinder supplies the bounded parameter derivative used by the cited theorem; the whole $J_0$ need not be bounded. Fields that are merely integrable, motions that fail injectivity somewhere, and time intervals whose closure meets the boundary of $I$ are outside this statement.

• Reading of the two terms. At a fixed spatial point ${\mathrm{\partial }}_{t}f$ is the local rate of change of the field; $u\cdot \mathrm{\nabla }f$ is the transport of the field by the material velocity. Both have dimension $[f]{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), so the display is dimensionally consistent, and the singular term $f\mathrm{div}u$ records the change of volume of a material element along the motion.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(B,O,I)$ be body data (Continuum body data, the fixed inertial frame and the dimensions of the fields), let $\chi$ be a motion of the model with spatial velocity $u$ (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum motions, spatial velocity, material acceleration and swept regions), let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$, let $\mathrm{\Sigma }$ be the swept region over $\overline{J_0}$, and let $\rho :U\to (0,\mathrm{\infty })$ be a density field of class $C^1$ on an open set $U$ containing $\mathrm{\Sigma }$ (The density field of the continuum model and the mass of a material subregion, $C^k$ Euclidean maps and diffeomorphisms). Assume the mass-balance postulate (CM) of Adopted mass balance: the mass of every material subregion is constant in time for every material subregion and every $t\in J_0$. Then:

(i) Local form. For every $(t,x)\in J_0\times {\chi }_t(B)$,

$$\begin{array}{cccc}& {\mathrm{\partial }}_t\rho (t,x)+\mathrm{div}\left(\rho u\right)(t,x)=0.& & \text{(C)}\end{array}$$

Conversely, if (C) holds on an open neighbourhood of $\mathrm{\Sigma }$, then (CM) holds for every material subregion and every $t\in J_0$.

(ii) Material form. On $J_0\times {\chi }_t(B)$ the material derivative of the density satisfies

$D_t\rho =-\rho \mathrm{div}u,$

and writing $J(t,X)=\det D_X{\chi }_t(X)$ along the motion, the product $\rho (t,\chi (t,X))J(t,X)$ is constant in $t$ for every label $X\in B$.

(iii) Volume rate. For every $X\in B$ and $t\in J_0$ one has ${\mathrm{\partial }}_{t}J(t,X)=J(t,X)\mathrm{div}u(t,\chi (t,X))$: the relative rate of change of an infinitesimal material volume is $\mathrm{div}u$, and the identity of (ii) is the statement that density times volume is conserved along a label. In particular $\mathrm{div}u=0$ on the swept region if and only if $\rho$ is constant along every label worldline ($D_t\rho =0$).

The statement is a conditional deduction from the adopted mass balance and the kinematic model; it asserts no equation of motion, no constitutive relation and no empirical value.

• Reading of the equation. The partial differential equation (C) is a balance of the model density with the mass flux $\rho u$; the material form $D_t\rho =-\rho \mathrm{div}u$ says that density decreases along a label exactly when the material element dilates. The identity $\rho J=\text{constant}$ along a label is the precise sense in which mass is conserved in the model, and it is the form used by the wave and fluid items below.

• Conditionality. The equation is a theorem about the adopted model: it holds because the model is assumed to balance mass, and it fails if the mass-balance assumption is replaced by a balance with sources. No claim is made that any material's density satisfies it beyond the modelling regime of Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance.

Fix body data $(B,O,I)$ and a motion $\chi$ of the model with current regions ${\mathrm{\Omega }}_t={\chi }_t(B)$ and $\mathcal{U}=\{(s,x):s\in I,x\in {\chi }_s(O)\}$. Let $U\subseteq \mathcal{U}$ be open and contain the swept closure $\{(s,x):s\in \overline{J_0},x\in {\chi }_s(\bar{B})\}$, where $J_0$ is an open interval with compact closure in $I$ (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum motions, spatial velocity, material acceleration and swept regions). Use on $U$ the restriction of the positive $C^1$ density slot $\rho :\mathcal{U}\to (0,\mathrm{\infty })$ of Continuum body data, the fixed inertial frame and the dimensions of the fields, and denote that restriction again by $\rho$. Let $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\parallel }n{\mathrm{\parallel }}_2=1\}$ be the unit sphere.

A traction field of the model is a map

$t:U\times S^2⟶{\mathbb{R}}^3,$

read with the convention of Traction fields and stress matrices representing them: $t(s,x,n)$ is the force per unit area exerted by the material on the side towards which $n$ points upon the material on the other side, at the point $x$ of a surface whose unit normal is $n$, at time $s$. Its dimension is that of force per area, $[t]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$, read in $\mathrm{N}{\mathrm{m}}^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

A $C^1$ field of matrices $\sigma :U\to M_3(\mathbb{R})$ represents the traction field when

$t(s,x,n)=\sigma (s,x)n\text{for every }(s,x,n)\in U\times S^2,$

the product being matrix-vector multiplication. The existence of such a $\sigma$ for a given traction field is not asserted here: it is a conditional conclusion of the balance items of this page. When it exists, $\sigma (s,x)$ is the stress matrix of the model at that event, and its entries read ${\sigma }_{ij}=(t(s,x,e_j){)}_i$: the $i$-th component of the traction across the surface with normal $e_j$.

The isotropic matrix $-\lambda I_3$ with $\lambda \in \mathbb{R}$ represents the normal traction $t(n)=-\lambda n$: under the convention above, positive $\lambda$ is a compressive normal traction and $\lambda$ is called the pressure in this sign convention. Within this isotropic family, $\lambda =0$ gives the zero stress matrix, a pressure-free unstressed state. The body force per unit mass is a separate object: a continuous field $b:U\to {\mathbb{R}}^3$ with the dimension of acceleration, $[b]=\mathrm{\ell }{\tau }^{-2}$. The body-force density per unit volume is $\rho b:U\to {\mathbb{R}}^3$, of dimension $\mu {\mathrm{\ell }}^{-2}{\tau }^{-2}$. A traction field need not satisfy $t(-n)=-t(n)$ and its representing matrix, when one exists, need not be symmetric; both properties are separate conditional conclusions.

• Cauchy's hypothesis is a modelling assumption. The definition lets the traction depend on the surface only through its unit normal at the point. Contact interactions that depend on the curvature of the surface, on a surface couple distribution, or on a thin boundary layer would require a richer object; the counterexample item of the companion page shows that the global symmetry $t(-n)=-t(n)$ alone, without a balance law, does not even force the existence of a representing matrix.

• Units and interpretation. The values of $t$ and $\sigma$ are model quantities in the fixed frame of Continuum body data, the fixed inertial frame and the dimensions of the fields. A measured force per area is related to them only through a measurement model and is not asserted by this definition; no measured stress is recorded on this page.

• Sign convention. The convention is the one of Traction fields and stress matrices representing them: $t(n)$ acts on the material on the side opposite to that towards which $n$ points. With this convention a fluid whose total stress is assumed to lie in the isotropic family defined above has $\sigma =-p{I}_3$, so positive $p$ gives compression in the direction $-n$. Positive pressure alone does not impose this isotropic stress assumption.

Adopt for every model of the continuum kinematic branch (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance) the following description of the forces acting on the body, read in the fixed inertial frame and with the dimensions of Continuum body data, the fixed inertial frame and the dimensions of the fields.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Lebesgue volume and surface integrals below. Let $\rho$ be the positive $C^1$ density of The density field of the continuum model and the mass of a material subregion. Use a common open field domain $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ containing the full swept closure $\{(s,x):s\in I,x\in {\chi }_s(\bar{B})\}$. At time $s\in I$, write $U_s:=\{x:(s,x)\in U\}$ and ${\mathrm{\Omega }}_s:={\chi }_s(B)$. An admissible region at time $s$ is a nonempty bounded open $D\subseteq {\mathrm{\Omega }}_s$ with $\bar{D}\subseteq U_s$ and a specified finite piecewise $C^1$ boundary presentation in Specified finite piecewise C1 boundary presentations. Its faces $S_j$ carry the outward unit normal ${\nu }_D$ off the surface-null edge set. Boundary integration means the finite sum of the face integrals of Surface integration on compact C1 hypersurfaces, componentwise for vectors; volume integration uses the Lebesgue integral over $D$ (Integral over a measurable subset), with signed integrals defined by positive and negative parts in Integrable real and complex functions, and their integrals. Compact closure, continuous fields and finite face areas make these integrals finite.

(i) Body forces. There is a continuous field $b:U\to {\mathbb{R}}^3$ on the common open field domain $U$, the body force per unit mass, of dimension $[b]=\mathrm{\ell }{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The body force exerted on an admissible region $D$ at time $s$ is ${\int }_D\rho (s,x)b(s,x)dx$, of dimension $\mu \mathrm{\ell }{\tau }^{-2}$ (force).

(ii) Contact forces and Cauchy's hypothesis. There is a traction field $t:U\times S^2\to {\mathbb{R}}^3$ of class $C^1$, hence continuous, in the sense of Traction fields and stress matrices of the continuum model, with units and sign conventions, force per unit area, with dimension $[t]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$. The definition of a traction field already encodes Cauchy's hypothesis: the contact force exerted across a surface at a point depends on the surface only through the value of its unit normal at that point, so two surfaces through the same point with the same unit normal experience the same traction. The contact force exerted across the boundary of an admissible region $D$ at time $s$ is ${\int }_{\mathrm{\partial }D}t(s,y,{\nu }_D(y))dS(y)$, with the outward unit normal ${\nu }_D$ and the finite face-sum integral defined above. No dependence of $t$ on the curvature of the surface, on higher derivatives of the surface, or on the distant shape of the body is introduced; no surface couples, no traction singularities and no contact forces on curves are introduced.

(iii) Exclusions. This force description is not adopted, and no statement of this page covers: couple stresses or distributions of surface couples; contact interactions that depend on the geometry of the surface beyond its tangent plane at the point (thin boundary layers, capillarity at contact lines, adhesion laws with curvature dependence); nonlocal contact interactions; electromagnetic, thermal, chemical and radiative forces; and body couples. A model with such features requires an extended force description and its own balance laws.

(iv) Status. These are adopted assumptions about how forces are represented in the model. No balance of forces is asserted by this postulate; the balance principles are the subject of the next postulate, and the existence of a representing stress matrix is a conditional conclusion of the theorem that applies Cauchy's tetrahedron argument to those balances. The fields $b$ and $t$ are model quantities; no measured force is asserted, and the frame-covariance requirement of the kinematic branch applies to them.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Lebesgue volume and surface integration used in these general balances. The pointwise field and open-region predicates are unaffected.

Adopt for every model of the continuum branch, together with the force description of Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions and the fields of The density field of the continuum model and the mass of a material subregion and Traction fields and stress matrices of the continuum model, with units and sign conventions, the following balance principles.

(i) Setting. Let $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ be an open set containing the swept region of the motion over its time window. The density $\rho$, the spatial velocity $u$ and the material acceleration $a=D_{t}u$ are continuous on $U$ (Euclidean spatial calculus: gradient, divergence and material derivative, Continuum motions, spatial velocity, material acceleration and swept regions), and $b$ and $t$ are the body-force and traction fields of Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions. For $s$ in the time window the current region is ${\mathrm{\Omega }}_s={\chi }_s(B)$.

(ii) Admissible regions. An admissible region at time $s$ is a bounded piecewise $C^1$ Euclidean domain with a specified finite presentation with $D\subseteq {\mathrm{\Omega }}_s$, whose closure lies in the open domain of the fields at time $s$ (Specified finite piecewise C1 boundary presentations), with outward unit normal ${\nu }_D$ off its edge set and boundary term the sum of the surface integrals over its faces (Surface integration on compact C1 hypersurfaces).

(iii) Balance of forces. For every time $s$ and every admissible region $D$ at time $s$,

$$\begin{array}{cccc}& {\int }_{\mathrm{\partial }D}t(s,y,{\nu }_D(y))dS(y)+{\int }_D(\rho b-\rho a)(s,y)dy=0.& & \text{(F)}\end{array}$$

Read as a momentum balance, (F) says that the resultant contact force on $D$ equals ${\int }_D\rho ady-{\int }_D\rho bdy$; equivalently contact plus body force equals ${\int }_D\rho ady$, i.e. the total applied force equals the mass times acceleration of the region, with the inertial term written as a volume force.

(iv) Balance of torques, with no couples. For every time $s$, every admissible region $D$ at time $s$, and every fixed origin $x_0\in {\mathbb{R}}^3$,

$$\begin{array}{cccc}& {\int }_{\mathrm{\partial }D}(y-x_0)\times t(s,y,{\nu }_D(y))dS(y)+{\int }_D(y-x_0)\times (\rho b-\rho a)(s,y)dy=0,& & \text{(T)}\end{array}$$

the cross product being that of ${\mathbb{R}}^3$ (The cross product in ${\mathbb{R}}^3$). The balance is imposed with the same volume force density $\rho b-\rho a$ as in (F); there are no body couples and no surface couples. Since the identity holds for every origin, the choice of $x_0$ is immaterial given (F).