# time dilatation of a volume element

## Formulae derivation

Let $X^{\alpha}$ and $x^{i}$ be material and spatial coordinates, respectively. Consider diffeomorphic the mapping 1

 $\displaystyle X^{\alpha}\mapsto x^{i}(X^{\alpha},\tau),\qquad t_{0}\leq\tau% \leq{t},$

which represents the motion of continuum $\Re\subset(\mathbb{R}^{3},\lVert\cdot\rVert)$ . The Jacobian   $J=|x^{i}_{\;,\alpha}|$ (comma denoting partial differentiation with respect to the indicated coordinate   ) of coordinate transformation  is given by

 $\displaystyle J=\epsilon^{\alpha\beta\gamma}\;x^{1}_{\;,\alpha}\,x^{2}_{\;,% \beta}\,x^{3}_{\;,\gamma}\equiv C^{\gamma}_{3}\;x^{3}_{\;,\gamma}\;,$

$\epsilon^{\alpha\beta\gamma}$ being the Levi-Civita density, $C^{\gamma}_{3}$ the cofactor  of $x^{3}_{\;,\gamma}$ in the determinant  expansion and it comes expressed as

 $\displaystyle C^{\gamma}_{3}=\epsilon^{\alpha\beta\gamma}\;x^{1}_{\;,\alpha}\,% x^{2}_{\;,\beta}=\frac{1}{2}(\epsilon^{\alpha\beta\gamma}\;x^{1}_{\;,\alpha}\,% x^{2}_{\;,\beta}+\epsilon^{\beta\alpha\gamma}\;x^{1}_{\;,\beta}\,x^{2}_{\;,% \alpha})$
 $\displaystyle=\frac{1}{2}\epsilon^{\alpha\beta\gamma}\;(x^{1}_{\;,\alpha}\,x^{% 2}_{\;,\beta}-x^{1}_{\;,\beta}\,x^{2}_{\;,\alpha})=\frac{1}{2}\epsilon^{\alpha% \beta\gamma}\epsilon_{ij3}\;x^{i}_{\;,\alpha}\;x^{j}_{\;,\beta}.$

So that, for any arbitrary cofactor $C^{\gamma}_{k},$

 $\displaystyle C^{\gamma}_{k}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}\epsilon_{% ijk}\;x^{i}_{\;,\alpha}\;x^{j}_{\;,\beta}.$

Let us multiply by $x^{n}_{\;,\gamma}.$

 $\displaystyle C^{\gamma}_{k}\;x^{n}_{\;,\gamma}=\frac{1}{2}\epsilon_{ijk}\;% \epsilon^{\alpha\beta\gamma}\;x^{i}_{\;,\alpha}\;x^{j}_{\;,\beta}\;x^{n}_{\;,% \gamma}=\frac{1}{2}|{x}^{i}_{\;,\alpha}|\epsilon_{ijk}\;\epsilon^{ijn}=J\delta% ^{n}_{k},\qquad\frac{C^{\gamma}_{k}}{J}\;x^{n}_{\;,\gamma}=\delta^{n}_{k},$ (1)

where we have used well-known alternator’s properties. Moreover, since the cofactor $C^{\gamma}_{k}$ is independent on $x^{k}_{\;,\gamma}$ by its own definition,

 $\displaystyle\frac{\partial}{\partial{x^{n}_{\;,\gamma}}}(J\delta^{n}_{k})=% \frac{\partial{J}}{\partial{x^{k}_{\;,\gamma}}}=C^{\gamma}_{k}.$ (2)

But

 $\displaystyle X^{\gamma}_{\;,k}\;x^{n}_{\;,\gamma}=\delta^{n}_{k},$

which it is compared with Eq.(1) to obtain

 $\displaystyle X^{\gamma}_{\;,k}=\frac{C^{\gamma}_{k}}{J},\qquad C^{\gamma}_{k}% =JX^{\gamma}_{\;,k}.$

So from Eq.(2) we get

 $\displaystyle\frac{\partial{J}}{\partial{x^{k}_{\;,\gamma}}}=JX^{\gamma}_{\;,k}.$ (3)

Let us now consider the relation  $dv=JdV$ between the spatial and material volume elements, and by taking the material time derivative

 $\displaystyle\dot{\overline{dv}}=\dot{J}dV,$ (4)

because $\dot{\overline{dV}}=0,$ by definition. From Eqs.(3)-(4),

 $\displaystyle\dot{\overline{\Big{(}\frac{dv}{dV}\Big{)}}}=\frac{\partial{J}}{% \partial{x^{i}_{\;,\alpha}}}\;\dot{\overline{x^{i}_{\;,\alpha}}}=\frac{% \partial{J}}{\partial{x^{i}_{\;,\alpha}}}\;\dot{x}^{i}_{\;,\alpha}=JX^{\alpha}% _{\;,i}\;v^{i}_{\;,\alpha}=Jv^{i}_{\;,i}$

where $\dot{x}^{i}_{\;,\alpha}\equiv{v^{i}_{\;,\alpha}}$ are material gradient  components of velocity field, thus arriving to the result due to Euler

 $\displaystyle\frac{\dot{J}}{J}=v^{i}_{\;,i}\equiv\nabla_{x}\cdot\mathbf{v},% \qquad\dot{\overline{\log{J}}}=\nabla_{x}\cdot\mathbf{v},$ (5)

expressing the spatial divergence  of velocity field. Also, by substituting $dV=dv/J$ in Eq.(4) we get

 $\displaystyle\dot{\overline{\log{dv}}}=\nabla_{x}\cdot\mathbf{v}.$ (6)

## Physical interpretations

• The time logarithm of dilatation and the first invariant $I_{\nabla_{x}\mathbf{v}}$ associate to the tensor of velocity spatial gradient, coincide exactly.

• If we consider the Lagrangian strain tensor $E_{ij}=1/2(u_{i,j}+u_{j,i}+u_{i,k}u_{k,j})$ (large strain) for $small"$ strain, i.e. the initial undistorsioned (material) reference configuration  $\chi_{\varkappa}(X_{i},t_{0})$ maps to near distorsioned (spatial) reference configuration $\chi_{\varkappa+\Delta\varkappa}(X_{i},\tau)$ (as $\tau\to t_{0}$) during the motion of continuum $\Re,$ coinciding approximately the spatial coordinates with the material coordinates and therefore, as a consequence, the quadratic displacement gradient $u_{i,k}u_{k,j}\approx 0.$ 22Indeed for small strain is required that $\nabla_{X}\mathbf{u}\cdot\mathbf{u}\nabla_{X}\approx\mathbf{0}$ and $\nabla_{X}\mathbf{u}\cdot\nabla_{x}\mathbf{u}\approx\mathbf{0}.$ To see this, we consider the coordinates transformation $\mathbf{x}(\tau)=\chi(\mathbf{X},\tau),$ as $\tau\to{t_{0}}.$ So $d\mathbf{x}=d\mathbf{X}\cdot\nabla_{X}\mathbf{x}.$ But $d\mathbf{u}=d\mathbf{X}\cdot\nabla_{X}\mathbf{u}=d\mathbf{x}\cdot\nabla_{x}% \mathbf{u},$ then by the first equation, we have $d\mathbf{u}=(d\mathbf{X}\cdot\nabla_{X}\mathbf{x})\cdot\nabla_{x}\mathbf{u}=d% \mathbf{X}\cdot(\nabla_{X}\mathbf{x}\cdot\nabla_{x}\mathbf{u}),$ and since $d\mathbf{X}$ is arbitrary $\nabla_{X}\mathbf{u}=\nabla_{X}\mathbf{x}\cdot\nabla_{x}\mathbf{u}.$ (The chain rule  !) Recalling now $\mathbf{x}=\mathbf{X}+\mathbf{u},$ we get $\nabla_{X}\mathbf{u}=[\nabla_{X}(\mathbf{X}+\mathbf{u})]\cdot\nabla_{x}\mathbf% {u}=(\mathbf{1}+\nabla_{X}\mathbf{u})\cdot\nabla_{x}\mathbf{u}=\nabla_{x}% \mathbf{u}+\nabla_{X}\mathbf{u}\cdot\nabla_{x}\mathbf{u},$ which shows that quadratic gradient is approximately equal to zero whenever $\nabla_{X}\mathbf{u}\approx\nabla_{x}\mathbf{u},$ i.e. the material undistorsioned reference configuration $\varkappa$ be approximately equal to the spatial distorsioned configuration $\varkappa+\Delta\varkappa.$ In fact elasticity theory defines tensor as $e_{ij}\equiv\ 1/2(u_{i,j}+u_{j,i}),$ i.e. $e_{ij}\approx E_{ij}$ for small strain, but the definition of tensor $\mathbf{e}$ is exact. So, in the vector displacement $u_{i}=x_{i}-X_{i},$ we take the material rate $\dot{u}_{i}=\dot{x}_{i}\equiv{v}_{i}$ and hence $\dot{u}_{i,j}=v_{i,j}.$ Therefore, according to the mentioned approximation, the material time derivative for tensors $\mathbf{E}$ and $\mathbf{e}$ are given by $\dot{E}_{ij}\approx\dot{e}_{ij}\approx 1/2(v_{i,j}+v_{j,i}).$ 33The last approximation because the tensors $\mathbf{E}$ and $\mathbf{e}$ are usually defined with respect to material coordinates and not with respect to the spatial ones. Now by contracting $j=i$, we get $\dot{E}_{ii}\approx\dot{e}_{i,i}\approx v_{i,i}=\nabla\cdot\mathbf{v}.$ 44Notice that although this is an approximated result, Eqs.(5)-(6) are exact.

• Considering the infinitesimal strain tensor $\mathbf{e}$ (or $\mathbf{E}$ for small strain), we see that sum of normal strain $e_{ii}=e_{11}+e_{22}+e_{33}=u_{1,1}+u_{2,2}+u_{3,3}$ represents the trace or first invariant $I_{\mathbf{e}}$. Thus, in the initial undistorsioned reference configuration $\varkappa,$ we can use principal centered axes (i.e. along the eigenvectors    of tensor $\mathbf{e},$ whose representation corresponds to pure normal strains) of a volume element $dV$ in order to measure the induced dilatation $(dv-dV)/dV$ in the distorsioned reference configuration $\varkappa+\Delta\varkappa$. So, for an elemental rectangular parallelopiped of volume $dV$, we have

 $\displaystyle\Big{(}\frac{dv}{dV}\Big{)}\approx(1+e_{11})(1+e_{22})(1+e_{33})=% 1+e_{ii}+o(e^{2}_{nn}).\quad(n\;not\;summed)$

By taking now the material time derivative,

 $\displaystyle\dot{\overline{\Big{(}\frac{dv}{dV}\Big{)}}}\approx\dot{e}_{ii}% \approx\dot{E}_{ii}\approx\dot{u}_{i,i}=\dot{x}_{i,i}=v_{i,i}=\nabla\cdot% \mathbf{v},$

thus completing the aimed physical interpretation.

• A volume-preserving motion is said to be isochoric. Then

 $\displaystyle J=\bigg{|}\frac{\partial x^{i}}{\partial X^{\alpha}}\bigg{|}=1,% \qquad\operatorname{div}_{x}\mathbf{v}=0.$

$V$ is called material volume and $v$ is called control volume.

• Although we have used Cartesian rectangular systems, if we introduce generalizated tensors, all the results obtained are also valid for curvilinear coordinates . For instance,

1. (a)

Divergence

 $\displaystyle\nabla_{x}\cdot\mathbf{v}=\mathbf{g}^{i}\cdot\frac{\partial}{% \partial{x^{i}}}(v^{j}\mathbf{g}_{j})=\mathbf{g}^{i}\cdot\mathbf{g}_{j}v^{j}|_% {i}=v^{i}|_{i}=v^{i}_{\;,i}+\Gamma^{j}_{ji}v^{i},$
 $\displaystyle\Gamma^{j}_{ji}=\frac{1}{2g}\frac{\partial{g}}{\partial{x}^{i}},% \quad g=|{g_{ij}}|,\quad g_{jk}g^{ki}=\delta^{i}_{j},$

where $v^{j}|_{i},\,\mathbf{g}^{i},\,\mathbf{g}_{j},$ stand for covariant derivative  and contravariant and covariant spatial base vectors, respectively.

2. (b)

 $\displaystyle\nabla_{X}\mathbf{u}=\mathbf{G}^{\alpha}\frac{\partial}{\partial{% X}^{\alpha}}(u^{\beta}\mathbf{G}_{\beta})=u^{\beta}|_{\alpha}\mathbf{G}^{% \alpha}\mathbf{G}_{\beta}=(u^{\beta}_{\;,\alpha}+\Gamma^{\beta}_{\gamma\alpha}% u^{\gamma})\mathbf{G}^{\alpha}\mathbf{G}_{\beta},$
 $\displaystyle\Gamma^{\beta}_{\gamma\alpha}=G^{\beta\delta}\Gamma_{\gamma\alpha% \delta},\quad\Gamma_{\gamma\alpha\delta}=\frac{1}{2}(G_{\gamma\delta,\alpha}+G% _{\alpha\delta,\gamma}-G_{\gamma\alpha,\delta}).$
 $\displaystyle\mathbf{u}\nabla_{X}=\frac{\partial}{\partial{X}^{\beta}}(u^{% \alpha}\mathbf{G}_{\alpha})\mathbf{G}^{\beta}=u^{\alpha}|_{\beta}\mathbf{G}_{% \alpha}\mathbf{G}^{\beta}=(u^{\alpha}_{\;,\beta}+\Gamma^{\alpha}_{\gamma\beta}% u^{\gamma})\mathbf{G}_{\alpha}\mathbf{G}^{\beta},$

where $\mathbf{G}_{\alpha}\;,\,\mathbf{G}^{\beta}$ are the covariant and contravariant material base vectors, respectively. mutatis mutandis for spatial gradient tensors $\nabla_{x}\mathbf{u}\;,\,\mathbf{u}\nabla_{x}.$

3. (c)

Material time derivative

 $\displaystyle\dot{\mathbf{v}}=\dot{\overline{v^{i}\mathbf{g}_{i}}}=\frac{% \partial{v^{i}}}{\partial{t}}\mathbf{g}_{i}+v^{i}|_{j}v^{j}\mathbf{g}_{i}=% \frac{\partial{v^{i}}}{\partial{t}}\mathbf{g}_{i}+(v^{i}_{\;,j}v^{j}+\Gamma^{i% }_{kj}v^{k}v^{j})\mathbf{g}_{i},$

where the local time derivative  $\partial\mathbf{g}_{i}/\partial{t}=\mathbf{0},$ by definition.