Transformation of Some Quantities
Corotating Frame
The deformation gradient in the fixed frame is obtained by taking the derivative of Equation 3-1 with respect to X:
(3-10)
where Fr is the deformation gradient in the rotating frame.
Using this expression for the deformation gradient, the relation between the Green–Lagrange strain in the fixed and corotating frames is
A traction vector transforms between different frames like a vector. Therefore, the relation between the traction vector in the fixed and corotating frames is
where n = Rnr. Using Cauchy’s relation, t = σn, the Cauchy stress tensor transforms as
The first Piola–Kirchhoff tensor in the fixed frame is related to the same tensor in the corotating frame by
Space-Fixed Frame
Deformation gradient in space-fixed frame can be written as
FR is the deformation gradient using rotated coordinates xR as reference. Thus Green–Lagrange strain can be expressed as
We know that area transformation between spatial and material frame are related as
If we use this transformation for the case where the rotor only rigidly rotates without any deformation, then
Thus, dAR = dA0 and nR = RN.
The relation between Cauchy stress and second Piola–Kirchhoff stress is
Using the rigidly rotated state as a reference, the second Piola–Kirchhoff stress is
Therefore, relation between second Piola–Kirchhoff stress using initial configuration and rotated configuration as reference is
(3-11)
Following the similar steps, the relation between first Piola–Kirchhoff stress using initial and rotated configuration as reference is
(3-12)