Micromechanics of composites/Proof 8

Relation between axial vector and displacement edit

Let be a displacement field. The displacement gradient tensor is given by . Let the skew symmetric part of the displacement gradient tensor (infinitesimal rotation tensor) be

Let be the axial vector associated with the skew symmetric tensor . Show that


The axial vector of a skew-symmetric tensor satisfies the condition

for all vectors . In index notation (with respect to a Cartesian basis), we have

Since , we can write


Therefore, the relation between the components of and is

Multiplying both sides by , we get

Recall the identity


Using the above identity, we get


Now, the components of the tensor with respect to a Cartesian basis are given by

Therefore, we may write

Since the curl of a vector can be written in index notation as

we have

where indicates the -th component of the vector inside the square brackets.