On invariant and locking-free formulations for planar arbitrarily curved beams with Timoshenko-Ehrenfest beam model and peridynamic differential operator

Abstract

This study presents three formulations for the static, dynamic, and fracture simulations of planar arbitrarily curved beams using kinematic assumptions of Timoshenko-Ehrenfest beam model and peridynamic differential operator (PDDO). Displacements of the beam axis and rotation of the cross-section are considered as unknowns of the kinematic description. Two variants of the equations of motion are derived by means of the principle of virtual work, i.e., the first form is expressed in terms of cross-sectional stress resultants, whereas the second variant is written in teams of kinematic unknowns. PDDO is then incorporated into the two variants of the equations of motion and the principle of virtual work to convert them from differential to integral expressions. The driving force behind developing three formulations is to address the critical deficiency of literature, i.e., an invariant and locking-free PD beam formulation for the analysis of beams having complex geometry is not yet proposed. In this study, the invariant property of the proposed formulations is elucidated by theoretical means, and the locking effects are examined by numerical experiments. Several well-established examples are exhibited to assess the accuracy and robustness of the proposed formulations.