# Strain-Displacement relationship symmetrization

In the context of infinitesimal elastic strain theory, one writes the relationship between displacement and strain as

$$\epsilon_{ij} = \frac{1}{2}( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} )$$

My question is; is the full non-simmetrized displacement tensor a meaningful mechanical quantity? what is the physical interpretation of the anti-simmetric part of the displacement gradient, that is:

$$\omega_{ij}=\frac{1}{2}( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} )$$

In what circumstances does this physical quantity plays a role in the structural analysis of the stresses in a material?

• If you wish to rotate your principal axis around, you will find it much easier to do with the real, physically meaningful tensor formalism. I've never liked the reduced engineering format - I think it is a relic from the dinosaur ages (much like secant and cosecant for that matter). Oct 21, 2014 at 17:36
• can you point a reference for this and elaborate a bit? Oct 21, 2014 at 17:59
• Most engineering elasticity texts end up with $\sigma_{i} = c_{i,j}\epsilon_{j}$, using $i$ and $j$ = $x,y,z,xy,xz,yz$ (only 6 elements) and throwing in the appropriate factors of 2 or 1/2 on the off-diagonal terms to make it work out right. That is, they assume that $xy = yx$ and shove it into one index. Physics folk would recognize this better as $\sigma_{ij} = c_{ij,kl}\epsilon_{kl}$. This eliminates the need for the seemingly random factors of 2 or 1/2 on the off-diagonal elements, and makes it straightforward to use Euler formalism to rotate the axes to what you'd like them to be. Oct 21, 2014 at 18:53
• you are referring to the Voigt notation: en.wikipedia.org/wiki/Voigt_notation Oct 21, 2014 at 19:10
• but in this case the anti-symmetric components of displacements are in principle invariant under such relabeling or axis reorientation. The anti-symmetric component of a 3x3 matrix transforms as an axial vector Oct 21, 2014 at 19:12