3.2. Extracting the axis and angle of local rotation

Try these first

1. 1.

   Define the axial vector and its significance in rotation.

   Hint

   Think about how direction relates to rotation.

2. 2.

   What does a skew-symmetric tensor indicate?

   Hint

   Consider its structure – zero diagonals.

3. 3.

   What represents the magnitude of the axial vector?

   • Angle of rotation
   • Axis of rotation
   • Skew-symmetric tensor

   Hint

   Think about what an angle represents in rotations.

4. 4.

   True or False: The skew-symmetric tensor can only represent a single axis of rotation.

   • True
   • False

   Hint

   Consider how many planes can exist in 3D.

5. 5.

   Given a skew-symmetric tensor: W = [ 0 -1 2; 1 0 -0; -2 0 0], derive the axial vector and the angle of rotation.

   Hint

   Start by identifying non-zero elements in W.

6. 6.

   In a material experiencing shear and local rigid rotation simultaneously, calculate the isolated effect of rotation on the angle given an axial vector representation.

   Hint

   Visualize how rotation happens even while shear is present.