Abstract In this paper, an analysis of kinematics of the isotropic elastic Cosserat continuum is presented in infinitesimal and finite deformations. Emphasis is given on the applicability of corotational stress rates for hypoelasticity in micro-polar continua. A non-linear finite element analysis is performed with an explicit formulation of tangent stiffness matrices in the case of Truesdell stress and couple stress rates. A comprehensive path-dependent procedure is employed based on the arc-length method to calculate the stability points and handle the snap-back problem. Finally, the accuracy and efficiency of method are illustrated by numerical examples.