A Palatini Variational Formulation of Cosserat Elasticity

Abstract

We present a Palatini-type geometric formulation of classical Cosserat elasticity in which the coframe and rotational connection are treated as independent variational fields. In contrast to conventional metric-based approaches, this formulation makes the underlying geometric structure explicit and separates translational and rotational degrees of freedom at the level of the action. The governing equations are obtained directly as Euler--Lagrange equations and yield the Cosserat force and moment balance laws without imposing compatibility constraints a priori. It is further shown that these balances arise naturally from invariance of the action under spatial translations and rotations via Noether's first theorem, providing a transparent variational interpretation of micropolar mechanics. A metric-free linearization recovers the classical strain and wryness measures and establishes equivalence with standard tensorial formulations under appropriate constitutive assumptions. The proposed framework clarifies the role of the connection field, which remains implicit in classical theories, and provides a unified geometric setting for Cosserat continua. In addition, it establishes the foundation for subsequent extensions in which torsion and curvature represent evolving defect densities and lead to mesoscopic theories of defect mechanics.