Geometrically non-linear 3D finite-element analysis of micropolar continuum

In this work a 3D geometrically non-linear micropolar finite element formulation based on Biot-like tensor representation of stresses, couple stresses, strains and curvature is presented. The discrete approximation is based on hexahedral finite elements with Lagrange interpolation for displacements and rotational spins. The complete residual derivation, linearisation and update are presented in detail. The elements are tested against a non-linear generalisation of the linear micropolar pure-bending problem, derived in this work, and it is shown that the elements converge to the derived solution. The elements are additionally tested on a combined bending and torsion problem and a well-known 45° bend as a genuine problem involving large 3D rotations, both also analysed in the framework of micropolar elasticity. It is observed that the derived finite elements are reliable and robust in modelling finite deformation problems in micropolar elasticity, exhibiting large displacements and large rotations and converge to reference geometrically non-linear classical elasticity solutions for small micropolar effects. With increasing micropolar effects the response becomes stiffer and the presented numerical examples may serve as benchmark problems to test geometrically non-linear micropolar finite elements including those involving large 3D rotations.