time dilatation of a volume element
So that, for any arbitrary cofactor
Let us multiply by
which it is compared with Eq.(1) to obtain
So from Eq.(2) we get
Let us now consider the relation between the spatial and material volume elements, and by taking the material time derivative
because by definition. From Eqs.(3)-(4),
expressing the spatial divergence of velocity field. Also, by substituting in Eq.(4) we get
The time logarithm of dilatation and the first invariant associate to the tensor of velocity spatial gradient, coincide exactly.
If we consider the Lagrangian strain tensor (large strain) for strain, i.e. the initial undistorsioned (material) reference configuration maps to near distorsioned (spatial) reference configuration (as ) during the motion of continuum coinciding approximately the spatial coordinates with the material coordinates and therefore, as a consequence, the quadratic displacement gradient 22Indeed for small strain is required that and To see this, we consider the coordinates transformation as So But then by the first equation, we have and since is arbitrary (The chain rule!) Recalling now we get which shows that quadratic gradient is approximately equal to zero whenever i.e. the material undistorsioned reference configuration be approximately equal to the spatial distorsioned configuration In fact elasticity theory defines infinitesimal strain tensor as i.e. for small strain, but the definition of tensor is exact. So, in the vector displacement we take the material rate and hence Therefore, according to the mentioned approximation, the material time derivative for tensors and are given by 33The last approximation because the tensors and are usually defined with respect to material coordinates and not with respect to the spatial ones. Now by contracting , we get 44Notice that although this is an approximated result, Eqs.(5)-(6) are exact.
Considering the infinitesimal strain tensor (or for small strain), we see that sum of normal strain represents the trace or first invariant . Thus, in the initial undistorsioned reference configuration we can use principal centered axes (i.e. along the eigenvectors of tensor whose representation corresponds to pure normal strains) of a volume element in order to measure the induced dilatation in the distorsioned reference configuration . So, for an elemental rectangular parallelopiped of volume , 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
is called material volume and is called control volume.
where stand for covariant derivative and contravariant and covariant spatial base vectors, respectively.
where are the covariant and contravariant material base vectors, respectively. mutatis mutandis for spatial gradient tensors
Material time derivative
where the local time derivative by definition.
- 1 L. Euler, Principes gÃÂ©nÃÂ©raux du mouvement des fluides, Hist. Acad. Berlin 1755, 274-315, 1757.
|Title||time dilatation of a volume element|
|Date of creation||2013-03-22 15:54:28|
|Last modified on||2013-03-22 15:54:28|
|Last modified by||perucho (2192)|