time dilatation of a volume element

Introduction

The rate of change of a volume element constitutes an important subject in certain applications related to the velocity field in fluid flow and elasticity, besides it admits some physical interpretations intimately entailed to tensor invariants, as we shall see.

Formulae derivation

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

$X^{\alpha }\mapsto x^i(X^{\alpha },\tau ),t_0\le \tau \le t,$

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

$J={ϵ}^{\alpha \beta \gamma }{x}_{,\alpha }^1{x}_{,\beta }^2{x}_{,\gamma }^3\equiv C_3^{\gamma }{x}_{,\gamma }^3,$

${ϵ}^{\alpha \beta \gamma }$ being the Levi-Civita density, $C_3^{\gamma }$ the cofactor of $x_{,\gamma }^3$ in the determinant expansion and it comes expressed as

$C_3^{\gamma }={ϵ}^{\alpha \beta \gamma }{x}_{,\alpha }^1{x}_{,\beta }^2={\displaystyle \frac{1}{2}}({ϵ}^{\alpha \beta \gamma }{x}_{,\alpha }^1{x}_{,\beta }^2+{ϵ}^{\beta \alpha \gamma }{x}_{,\beta }^1{x}_{,\alpha }^2)$

$={\displaystyle \frac{1}{2}}{ϵ}^{\alpha \beta \gamma }(x_{,\alpha }^1{x}_{,\beta }^2-x_{,\beta }^1{x}_{,\alpha }^2)={\displaystyle \frac{1}{2}}{ϵ}^{\alpha \beta \gamma }{ϵ}_{ij3}{x}_{,\alpha }^i{x}_{,\beta }^j.$

So that, for any arbitrary cofactor $C_k^{\gamma },$

$C_k^{\gamma }={\displaystyle \frac{1}{2}}{ϵ}^{\alpha \beta \gamma }{ϵ}_{ijk}{x}_{,\alpha }^i{x}_{,\beta }^j.$

Let us multiply by $x_{,\gamma }^n.$

$C_k^{\gamma }{x}_{,\gamma }^n={\displaystyle \frac{1}{2}}{ϵ}_{ijk}{ϵ}^{\alpha \beta \gamma }{x}_{,\alpha }^i{x}_{,\beta }^j{x}_{,\gamma }^n={\displaystyle \frac{1}{2}}|x_{,\alpha }^i|{ϵ}_{ijk}{ϵ}^{ijn}=J{\delta }_k^n,{\displaystyle \frac{C_k^{\gamma }}{J}}{x}_{,\gamma }^n={\delta }_k^n,$ (1)

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

${\displaystyle \frac{\partial }{\partial x_{,\gamma }^n}}(J{\delta }_k^n)={\displaystyle \frac{\partial J}{\partial x_{,\gamma }^k}}=C_k^{\gamma }.$ (2)

But

$X_{,k}^{\gamma }{x}_{,\gamma }^n={\delta }_k^n,$

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

$X_{,k}^{\gamma }={\displaystyle \frac{C_k^{\gamma }}{J}},C_k^{\gamma }=J{X}_{,k}^{\gamma }.$

So from Eq.(2) we get

${\displaystyle \frac{\partial J}{\partial x_{,\gamma }^k}}=J{X}_{,k}^{\gamma }.$ (3)

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

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

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

$\dot{\overline{\left({\displaystyle \frac{dv}{dV}}\right)}}={\displaystyle \frac{\partial J}{\partial x_{,\alpha }^i}}\dot{\overline{x_{,\alpha }^i}}={\displaystyle \frac{\partial J}{\partial x_{,\alpha }^i}}{\dot{x}}_{,\alpha }^i=J{X}_{,i}^{\alpha }{v}_{,\alpha }^i=J{v}_{,i}^i$

where ${\dot{x}}_{,\alpha }^i\equiv v_{,\alpha }^i$ are material gradient components of velocity field, thus arriving to the result due to Euler[1]

${\displaystyle \frac{\dot{J}}{J}}=v_{,i}^i\equiv {\nabla }_x\cdot 𝐯,\dot{\overline{\log J}}={\nabla }_x\cdot 𝐯,$ (5)

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

$\dot{\overline{\log dv}}={\nabla }_x\cdot 𝐯.$ (6)

Physical interpretations

• The time logarithm of dilatation and the first invariant $I_{{\nabla }_x𝐯}$ 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 strain, i.e. the initial undistorsioned (material) reference configuration ${\chi }_{\varkappa }(X_i,t_0)$ maps to near distorsioned (spatial) reference configuration ${\chi }_{\varkappa +\mathrm{\Delta }\varkappa }(X_i,\tau )$ (as $\tau \to t_0$) during the motion of continuum $\mathrm{\Re },$ coinciding approximately the spatial coordinates with the material coordinates and therefore, as a consequence, the quadratic displacement gradient $u_{i,k}{u}_{k,j}\approx 0.$ In fact elasticity theory defines infinitesimal strain tensor as $e_{ij}\equiv \mathrm{\hspace{0.25em}1}/2(u_{i,j}+u_{j,i}),$ i.e. $e_{ij}\approx E_{ij}$ for small strain, but the definition of tensor $𝐞$ 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 $𝐄$ and $𝐞$ are given by ${\dot{E}}_{ij}\approx {\dot{e}}_{ij}\approx 1/2(v_{i,j}+v_{j,i}).$ Now by contracting $j=i$, we get ${\dot{E}}_{ii}\approx {\dot{e}}_{i,i}\approx v_{i,i}=\nabla \cdot 𝐯.$

• Considering the infinitesimal strain tensor $𝐞$ (or $𝐄$ 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 $I_{𝐞}$. Thus, in the initial undistorsioned reference configuration $\varkappa ,$ we can use principal centered axes (i.e. along the eigenvectors of tensor $𝐞,$ 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 $\varkappa +\mathrm{\Delta }\varkappa$. So, for an elemental rectangular parallelopiped of volume $dV$, we have

  $\left({\displaystyle \frac{dv}{dV}}\right)\approx (1+e_{11})(1+e_{22})(1+e_{33})=1+e_{ii}+o(e_{nn}^2).\mathit{\hspace{1em}}(nnotsummed)$

  By taking now the material time derivative,

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

  thus completing the aimed physical interpretation.

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

  $J=\left|{\displaystyle \frac{\partial x^i}{\partial X^{\alpha }}}\right|=1,{\mathrm{div}}_x𝐯=0.$

  $V$ 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. Divergence

     ${\nabla }_x\cdot 𝐯={𝐠}^i\cdot {\displaystyle \frac{\partial }{\partial x^i}}(v^j{𝐠}_j)={{𝐠}^i\cdot {𝐠}_j{v}^j|}_i={v^i|}_i=v_{,i}^i+{\mathrm{\Gamma }}_{ji}^j{v}^i,$

     ${\mathrm{\Gamma }}_{ji}^j={\displaystyle \frac{1}{2g}}{\displaystyle \frac{\partial g}{\partial x^i}},g=|g_{ij}|,g_{jk}{g}^{ki}={\delta }_j^i,$

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

  2. Gradient

     ${\nabla }_X𝐮={𝐆}^{\alpha }{\displaystyle \frac{\partial }{\partial X^{\alpha }}}(u^{\beta }{𝐆}_{\beta })={u^{\beta }|}_{\alpha }{𝐆}^{\alpha }{𝐆}_{\beta }=(u_{,\alpha }^{\beta }+{\mathrm{\Gamma }}_{\gamma \alpha }^{\beta }{u}^{\gamma }){𝐆}^{\alpha }{𝐆}_{\beta },$

     ${\mathrm{\Gamma }}_{\gamma \alpha }^{\beta }=G^{\beta \delta }{\mathrm{\Gamma }}_{\gamma \alpha \delta },{\mathrm{\Gamma }}_{\gamma \alpha \delta }={\displaystyle \frac{1}{2}}(G_{\gamma \delta ,\alpha }+G_{\alpha \delta ,\gamma }-G_{\gamma \alpha ,\delta }).$

     $𝐮{\nabla }_X={\displaystyle \frac{\partial }{\partial X^{\beta }}}(u^{\alpha }{𝐆}_{\alpha }){𝐆}^{\beta }={u^{\alpha }|}_{\beta }{𝐆}_{\alpha }{𝐆}^{\beta }=(u_{,\beta }^{\alpha }+{\mathrm{\Gamma }}_{\gamma \beta }^{\alpha }{u}^{\gamma }){𝐆}_{\alpha }{𝐆}^{\beta },$

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

  3. Material time derivative

     $\dot{𝐯}=\dot{\overline{v^i{𝐠}_i}}={\displaystyle \frac{\partial v^i}{\partial t}}{𝐠}_i+{v^i|}_j{v}^j{𝐠}_i={\displaystyle \frac{\partial v^i}{\partial t}}{𝐠}_i+(v_{,j}^i{v}^j+{\mathrm{\Gamma }}_{kj}^i{v}^k{v}^j){𝐠}_i,$

     where the local time derivative $\partial {𝐠}_i/\partial t=\mathrm{𝟎},$ by definition.

Title	time dilatation of a volume element
Canonical name	TimeDilatationOfAVolumeElement
Date of creation	2013-03-22 15:54:28
Last modified on	2013-03-22 15:54:28
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	10
Author	perucho (2192)
Entry type	Definition
Classification	msc 53A45