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 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

Xαxi(Xα,τ),t0τt,

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

J=ϵαβγx,α1x,β2x,γ3C3γx,γ3,

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

C3γ=ϵαβγx,α1x,β2=12(ϵαβγx,α1x,β2+ϵβαγx,β1x,α2)
=12ϵαβγ(x,α1x,β2-x,β1x,α2)=12ϵαβγϵij3x,αix,βj.

So that, for any arbitrary cofactor Ckγ,

Ckγ=12ϵαβγϵijkx,αix,βj.

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)

But

X,kγx,γn=δkn,

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

X,kγ=CkγJ,Ckγ=JX,kγ.

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),

(dvdV)¯˙=Jx,αix,αi¯˙=Jx,αix˙,αi=JX,iαv,αi=Jv,ii

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

    (dvdV)(1+e11)(1+e22)(1+e33)=1+eii+o(enn2).(nnotsummed)

    By taking now the material time derivative,

    (dvdV)¯˙e˙iiE˙iiu˙i,i=x˙i,i=vi,i=𝐯,

    thus completing the aimed physical interpretation.

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

    J=|xiXα|=1,divx𝐯=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

      x𝐯=𝐠ixi(vj𝐠j)=𝐠i𝐠jvj|i=vi|i=v,ii+Γjijvi,
      Γjij=12ggxi,g=|gij|,gjkgki=δji,

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

    2. (b)

      Gradient

      X𝐮=𝐆αXα(uβ𝐆β)=uβ|α𝐆α𝐆β=(u,αβ+Γγαβuγ)𝐆α𝐆β,
      Γγαβ=GβδΓγαδ,Γγαδ=12(Gγδ,α+Gαδ,γ-Gγα,δ).
      𝐮X=Xβ(uα𝐆α)𝐆β=uα|β𝐆α𝐆β=(u,βα+Γγβαuγ)𝐆α𝐆β,

      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

      𝐯˙=vi𝐠i¯˙=vit𝐠i+vi|jvj𝐠i=vit𝐠i+(v,jivj+Γkjivkvj)𝐠i,

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

References

  • 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