time dilatation of a volume element


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 interpretationsMathworldPlanetmathPlanetmath intimately entailed to tensor invariantsMathworldPlanetmath, as we shall see.

Formulae derivation

Let Xα and xi be material and spatial coordinates, respectively. Consider diffeomorphic the mapping 11See motion of continuum


which represents the motion of continuum (3,) . The JacobianMathworldPlanetmathPlanetmath J=|x,αi| (comma denoting partial differentiation with respect to the indicated coordinateMathworldPlanetmathPlanetmath) of coordinate transformationMathworldPlanetmath is given by


ϵαβγ being the Levi-Civita density, C3γ the cofactorPlanetmathPlanetmath of x,γ3 in the determinantMathworldPlanetmath expansion and it comes expressed as


So that, for any arbitrary cofactor Ckγ,


Let us multiply by x,γn.

Ckγx,γn=12ϵijkϵαβγx,αix,βjx,γn=12|x,αi|ϵijkϵijn=Jδkn,CkγJx,γn=δkn, (1)

where we have used well-known alternator’s properties. Moreover, since the cofactor Ckγ is independent on x,γk by its own definition,

x,γn(Jδkn)=Jx,γk=Ckγ. (2)



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


So from Eq.(2) we get

Jx,γk=JX,kγ. (3)

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

dv¯˙=J˙dV, (4)

because dV¯˙=0, by definition. From Eqs.(3)-(4),


where x˙,αiv,αi are material gradientMathworldPlanetmath components of velocity field, thus arriving to the result due to Euler[1]

J˙J=v,iix𝐯,logJ¯˙=x𝐯, (5)

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

logdv¯˙=x𝐯. (6)

Physical interpretations

  • The time logarithm of dilatation and the first invariant Ix𝐯 associate to the tensor of velocity spatial gradient, coincide exactly.

  • If we consider the Lagrangian strain tensor Eij=1/2(ui,j+uj,i+ui,kuk,j) (large strain) for ``small" strain, i.e. the initial undistorsioned (material) reference configurationPlanetmathPlanetmath χϰ(Xi,t0) maps to near distorsioned (spatial) reference configuration χϰ+Δϰ(Xi,τ) (as τt0) during the motion of continuum , coinciding approximately the spatial coordinates with the material coordinates and therefore, as a consequence, the quadratic displacement gradient ui,kuk,j0. 22Indeed for small strain is required that X𝐮𝐮X𝟎 and X𝐮x𝐮𝟎. To see this, we consider the coordinates transformation 𝐱(τ)=χ(𝐗,τ), as τt0. So d𝐱=d𝐗X𝐱. But d𝐮=d𝐗X𝐮=d𝐱x𝐮, then by the first equation, we have d𝐮=(d𝐗X𝐱)x𝐮=d𝐗(X𝐱x𝐮), and since d𝐗 is arbitrary X𝐮=X𝐱x𝐮. (The chain ruleMathworldPlanetmath!) Recalling now 𝐱=𝐗+𝐮, we get X𝐮=[X(𝐗+𝐮)]x𝐮=(𝟏+X𝐮)x𝐮=x𝐮+X𝐮x𝐮, which shows that quadratic gradient is approximately equal to zero whenever X𝐮x𝐮, i.e. the material undistorsioned reference configuration ϰ be approximately equal to the spatial distorsioned configuration ϰ+Δϰ. In fact elasticity theory defines infinitesimalMathworldPlanetmathPlanetmath strain tensor as eij 1/2(ui,j+uj,i), i.e. eijEij for small strain, but the definition of tensor 𝐞 is exact. So, in the vector displacement ui=xi-Xi, we take the material rate u˙i=x˙ivi and hence u˙i,j=vi,j. Therefore, according to the mentioned approximation, the material time derivative for tensors 𝐄 and 𝐞 are given by E˙ije˙ij1/2(vi,j+vj,i). 33The last approximation because the tensors 𝐄 and 𝐞 are usually defined with respect to material coordinates and not with respect to the spatial ones. Now by contracting j=i, we get E˙iie˙i,ivi,i=𝐯. 44Notice that although this is an approximated result, Eqs.(5)-(6) are exact.

  • Considering the infinitesimal strain tensor 𝐞 (or 𝐄 for small strain), we see that sum of normal strain eii=e11+e22+e33=u1,1+u2,2+u3,3 represents the trace or first invariant I𝐞. Thus, in the initial undistorsioned reference configuration ϰ, we can use principal centered axes (i.e. along the eigenvectorsMathworldPlanetmathPlanetmathPlanetmath 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 ϰ+Δϰ. So, for an elemental rectangular parallelopiped of volume dV, we have


    By taking now the material time derivative,


    thus completing the aimed physical interpretation.

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


    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)



      where vj|i,𝐠i,𝐠j, stand for covariant derivativeMathworldPlanetmath and contravariant and covariant spatial base vectors, respectively.

    2. (b)



      where 𝐆α,𝐆β are the covariant and contravariant material base vectors, respectively. mutatis mutandis for spatial gradient tensors x𝐮,𝐮x.

    3. (c)

      Material time derivative


      where the local time derivativePlanetmathPlanetmath 𝐠i/t=𝟎, by definition.


  • 1 L. Euler, Principes généraux du mouvement des fluides, Hist. Acad. Berlin 1755, 274-315, 1757.
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