Derivation of Different Constitutive Laws for the Cosserat Medium

Abstract

In recent years, disciplines such as transport, space, and civil engineering are using lighter structural members to withstand the action of forces. On the one hand, this characteristic of structures results advantageous as smaller cross-sections are required, and therefore less material is used. On the other hand, the reduction of self-weight added into the design load-state makes these elements prone to suffering from strong undesired vibrations. Many different methods are addressed in the literature to mitigate the effect of unwanted vibrations: the installation of mass-tuned-dampers, the addition of viscoelastic materiales in the contact regions, friction dampers, or piezoelectric devices, among others. In this work the coupling mechanism for a passive vibration system in a beam-like structure modelled via the special theory of Cosserat rods is studied. The addition of a piezoelectric device in the mechanical structure is considered as a means to reducing the unwanted vibration phenomena and leads to a coupled electromechanical system. A procedure to derive the constitutive laws required for rod elements with mixed elastic material and piezoelectric devices is herein discussed. The present methodology could also be used to explore constitutive laws for different piezoelectric configurations, which could be of interest to control unwanted torsional vibrations in rotating structures such as drill-strings.