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Dilation Jacobian and advected scalar density

Example

Let $I \subseteq R$ be an open interval containing $0$ , let $U \subseteq R^{3}$ be nonempty and open, and let $α \in C^{2} (I, (0, \infty))$ satisfy $α (0) = 1$ . Define the material map $χ (t, a) = α (t) a$ for $a \in U$ , its inverse label $A (t, x) = x / α (t)$ for $x \in α (t) U$ , the derivative matrix $F = D_{a} χ$ , the Jacobian $J = \det F$ , and the Eulerian velocity $u (t, x) = \partial_{t} χ (t, A (t, x))$ . For any $ρ_{0} \in C^{1} (U, R)$ define $ρ (t, x) = α (t)^{- 3} ρ_{0} (x / α (t))$ . Here $C^{r}$ means continuous coordinate partial derivatives through total order $r$ , $D_{a} χ$ is the matrix of material-coordinate partial derivatives, $D_{x} u$ is the matrix of spatial coordinate partial derivatives, $tr M : = \sum_{i} M_{i i}$ for square matrices $M$ , and $I_{3}$ is the $3 \times 3$ identity matrix. Then

$F (t, a) = α (t) I_{3}, J (t, a) = α (t)^{3}, u (t, x) = \frac{α^{'} (t)}{α (t)} x, θ (t, x) : = tr (D_{x} u) = 3 \frac{α^{'} (t)}{α (t)}, ρ (t, χ (t, a)) J (t, a) = ρ_{0} (a) .$

The material pullback product $ρ (t, χ (t, a)) J (t, a)$ is independent of time and equals $ρ_{0} (a)$ . This is a mathematical kinematic example, not a physical model or measurement.

Example references

R. Abeyaratne, Continuum Mechanics, Volume II, §3.4.3 (printed pp. 71–72)

Facts & Assumptions

Given: $α$ , $U$ , $ρ_{0}$ , the dilation $χ (t, a) = α (t) a$ , and the formulas in the example statement.

[F1]

Smooth transport, Jacobian and advected reference density: under its stated smooth-motion and $C^{1}$ field hypotheses, the identity $ρ (t, χ (t, a)) J (t, a) = ρ (0, χ (0, a)) J (0, a)$ for every $a \in U$ and $t \in I$ is equivalent to the pointwise continuity equation; the lemma also characterizes it by constancy of integrals over every compact Jordan reference set.

[F2]

Set $Ω : = {(t, x) : t \in I, x \in α (t) U}$ . Since $U$ is nonempty and open, choose $a_{0} \in U$ and $ε > 0$ such that $B : = {a \in R^{3} : ∥ a - a_{0} ∥_{2} < ε}$ has $\overline{B} \subset U$ . This is a nonempty bounded open reference region with $C^{1}$ boundary. For the lemma's auxiliary field, take $q \equiv 0$ on $Ω$ ; this and the stated density $ρ$ are $C^{1}$ there. The map $χ (t, a) = α (t) a$ is jointly $C^{2}$ , and each slice is an orientation-preserving $C^{2}$ diffeomorphism with $J = α^{3} > 0$ . Thus the motion, reference-region, and field hypotheses of [F1] hold.

Verification

1.1givenalgebra

Differentiating $χ (t, a) = α (t) a$ in $a$ gives $F = α I_{3}$ . Since the space is three-dimensional, $\det F = α^{3}$ ; positivity of $α$ makes this an orientation-preserving dilation, and $α (0) = 1$ gives $F (0, a) = I_{3}$ and $J (0, a) = 1$ .

2.1givenstep 1.1algebra

The inverse label is $A = x / α$ . Differentiating $χ$ in time at fixed $a$ gives $\partial_{t} χ = α^{'} a = (α^{'} / α) x$ , so $u = (α^{'} / α) x$ ; taking its spatial derivative gives $D_{x} u = (α^{'} / α) I_{3}$ and $θ = 3 α^{'} / α$ .

2.2givenstep 1.1algebra

Substitution of $x = χ (t, a) = α (t) a$ into the stated density gives $ρ (t, χ (t, a)) = α^{- 3} ρ_{0} (a)$ . Multiplying by $J = α^{3}$ yields $ρ (t, χ (t, a)) J (t, a) = ρ_{0} (a)$ , which equals the initial pullback because $χ (0, a) = a$ and $J (0, a) = 1$ .

3.1F1F2step 2.2∎

The density is $C^{1}$ on the swept open set and $J > 0$ . Apply [F1] under the verified hypotheses in [F2] to the invariant product in step 2.2 to obtain the pointwise continuity equation and constant material integrals for every compact Jordan $K \subset U$ . This is a mathematical consequence of the displayed fields.

Depends on

Smooth transport, Jacobian and advected reference density

Dependency tree · 2 levels

Sources

R. Abeyaratne, Continuum Mechanics, Volume II of Lecture Notes on the Mechanics of Solids (MIT, 1988, revised 2020), §3.4.3