A Note on the Representation of Cosserat Rotation

This brief article provides an independent derivation of a formula given by Kafadar and Eringen (1971) connecting two distinct Cosserat spins. The first of these, the logarithmic spin represents the time rate of change of the vector defining finite Cosserat rotation, whereas the second, the instantaneous spin, gives the local angular velocity representing the infinitesimal generator of that rotation. While the formula of Kadafar and Eringen has since been identified by Iserles et al. (2000) as the differential of the Lie-group exponential, the present work provides an independent derivation based on quaternions. As such, it serves to bring together certain scattered results on quaternionic algebra, which is currently employed as a computational tool for representing rigid-body rotation in various branches of physics, structural and robotic dynamics, and computer graphics. 1 Background: Cosserat Rotations From the conventional continuum-mechanical viewpoint, a Cosserat continuum1 is defined via a differentiable map assigning spatial position x(x◦,t) and microstructural rotation P(x◦,t) to each material particle, with x = x◦, P = I in a given reference configuration, where P ∈ SO(3) denotes a real, proper orthogonal tensor. We can express the kinematics concisely in terms of the map R3 → R3 ×SO(3) given by x◦ → {x,θ}, where θ =− 2 ε :Θ , and Θ =−ε ·θ , (1) i.e. θi =−12εi jkΘ jk , and Θi j =−εi jkθ , J.D. Goddard Department of Mechanical and Aerospace Engineering 0411, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA, 92093-0411, USA e-mail: [email protected] 1 As noted by numerous authors, e.g. [1, 2, 3, 4], this is a special case of Eringen’s microstretch continuum [5], which involves both rotation and dilatation.