Continuum motions, spatial velocity, material acceleration and swept regions

Let $B\subseteq {\mathbb{R}}^3$ be a material region (nonempty, bounded, open, Material regions, densities and their mass) and let $I\subseteq \mathbb{R}$ be a nonempty open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

A motion of $B$ on $I$ consists of an open set $O\subseteq {\mathbb{R}}^3$ with $\bar{B}\subseteq O$ (Open ball, closed ball and sphere in a metric space) together with a map

$\chi :I\times O⟶{\mathbb{R}}^3$

such that:

1. $\chi$ is of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms);
2. for every $t\in I$ the map ${\chi }_t:=\chi (t,\cdot ):O\to {\mathbb{R}}^3$ is of class $C^1$, is injective, and has $D_X{\chi }_t(X)$ invertible for every $X\in O$;
3. writing

$J(t,X):=\det (D_X{\chi }_t(X))$

with the determinant of For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix, one has $J(t,X)>0$ for every $(t,X)\in I\times B$.

We call $X\in B$ the material label (reference position), $x=\chi (t,X)$ the position at time $t$ of the label $X$, and

${\mathrm{\Omega }}_t:={\chi }_t(B)$

the current (material) region at time $t$. Condition 2 makes each ${\chi }_t:O\to {\chi }_t(O)$ a $C^1$ diffeomorphism onto the open set ${\chi }_t(O)$ (An injective regular $C^k$ map is a $C^k$ diffeomorphism onto its image).

The spatial velocity field of the motion is

$u:\{(t,x):t\in I,x\in {\chi }_t(O)\}⟶{\mathbb{R}}^3,u(t,\chi (t,X)):={\mathrm{\partial }}_t\chi (t,X),$

and the spatial acceleration field is the material derivative of $u$ in the sense of Euclidean spatial calculus: gradient, divergence and material derivative,

$a(t,x):=D_{t}u(t,x)={\mathrm{\partial }}_{t}u(t,x)+Du(t,x)u(t,x)\mathrm{.}$

For a compact subinterval $K\subseteq I$ the swept region of $B$ over $K$ is

${\mathrm{\Sigma }}_K:=\{(t,x):t\in K,x\in {\chi }_t(\bar{B})\},$

and for a scalar field $f$ on a neighbourhood of ${\mathrm{\Sigma }}_K$ the spatial field $f(t,x)$ of a material quantity $F(t,X)$ is defined by $f(t,\chi (t,X))=F(t,X)$.

Put $\mathrm{\Phi }(t,X):=(t,\chi (t,X))$ on $I\times O$. By condition 1, $\mathrm{\Phi }$ is $C^1$ and its total derivative at $(t,X)$ is the block map $(s,Y)\mapsto (s,{\mathrm{\partial }}_t\chi (t,X)s+D_X{\chi }_t(X)Y)$, invertible because $D_X{\chi }_t(X)$ is invertible. If $\mathrm{\Phi }(t,X)=\mathrm{\Phi }(t^{\mathrm{\prime }},X^{\mathrm{\prime }})$, then $t=t^{\mathrm{\prime }}$ and injectivity of ${\chi }_t$ (condition 2) gives $X=X^{\mathrm{\prime }}$; so $\mathrm{\Phi }$ is injective and, by An injective regular $C^k$ map is a $C^k$ diffeomorphism onto its image, $\mathrm{\Phi }$ corestricts to a $C^1$ diffeomorphism from $I\times O$ onto the open set $\mathrm{\Phi }(I\times O)$. On that set the inverse map

$\mathrm{\Lambda }:={\mathrm{\Phi }}^{-1},\mathrm{\Lambda }(t,x)=(t,{\chi }_t^{-1}(x)),$

is $C^1$ by the same corollary. Hence $u(t,x)={\mathrm{\partial }}_t\chi (t,\mathrm{\Lambda }(t,x))$ is a composition of $C^1$ maps, so $u$ is $C^1$ on its open domain by the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$), and $a=D_{t}u$ is continuous there. Finally ${\mathrm{\Sigma }}_K=\mathrm{\Phi }(K\times \bar{B})$ is compact: it is the continuous image of a compact set.

• The positive Jacobian in condition 3 is required only on $I\times B$; the definition itself is purely mathematical and asserts neither conservation of mass nor any physical identification of $B$ with a body of matter. Physical readings (a constitutive law, a mass balance, a traction model) are separate adopted hypotheses in consuming items.

• Boundary regularity of $B$ is not part of this definition. Statements that integrate over ${\chi }_t(B)$ or $\mathrm{\partial }{\chi }_t(B)$ carry their own hypotheses; the transport theorem of this collection states the boundary regularity it uses, and the local balance statements say when ${\chi }_t(B)$ is required to be a bounded $C^1$ domain.

• The notation ${\chi }_t(O)$ keeps the domain of $u$ honest: $u(t,\cdot )$ is defined on the image of the whole reference chart $O$, not merely on ${\mathrm{\Omega }}_t$; this is what allows differentiation of $u$ at points of the current region with room to spare. Values of $u$ are spatial quantities, whereas ${\mathrm{\partial }}_t\chi$ is a material quantity; the identity $u(t,\chi (t,X))={\mathrm{\partial }}_t\chi (t,X)$ is the definition that relates them, and no equality of the two off the image is claimed.