Mechanics of Cosserat Media

We are trying to find out the work conjugate strain (and wryness) measures to T and M (stess and couple stress) and we know that in classical theory the conjugate strain measure is identified using the stress power, which appears in the power balance equation derived taking the inner product of the equation of motion with velocity. Please refer to the attached pdf for details.

1.  Given the equations of motion of Cosserat media (Eq 1 and 2 in attached pdf) how do we extend the procedure of calculating the stress power for Cosserat media?

This boiles down to the question of finding the time rate (material derivative) of the deformation map. Time rate of φ ( function that maps the position vectors from reference to spatial) is understood to be velocity v (Eq.1 in pdf). But how do we define the vector field ω (a material field) in Eq. 2 from Λ (function that maps reference directors to spatial)? Moreover, what would be the appropriate inner product to derive the power balance equation?

2. It is common to refer the relative velocity between two material points x and y as the difference between velocity vectors at x and y i.e. v(x)-v(y). We want a similar notion for relative ω (or relative material time rate of Λ), i.e. how can we define a quantity like ω(x)- ω(y)?