Large deformations of piezoelectric laminated beams based on the absolute nodal coordinate formulation

Piezoelectric beams can be deformed into many patterns and have potential application prospect in the field of smart foldable structures. Knowing the large deformation actuation mechanism of such structures is the basis in the soft machine design. In this paper, using the Equivalent Single-layer model, a new flexible piezoelectric laminated beam element based on the Absolute Nodal Coordinate Formulation (ANCF) is established. This beam element contains three layers, in which the base layer is made of linear elastic material, and the top and bottom layers are both made of piezoelectric materials. The constitutive equation of the piezoelectric material is introduced to derive the elastic force and the piezoelectric force. The correctness of the new beam element are validated by comparing with the analytical solution and the results from the equivalent bending moment equation. A group of large deformation patterns of the laminated beam are given to prove the applicability of the proposed element. Results show that when various voltages are applied to the piezoelectric actuators that located in arbitrary positions of the base beam, the flexible beam can achieve the desired shape smoothly. Dynamic responses of a highly flexible pendulum with and without piezoelectric layers are also investigated. Research in this work can be helpful to the understanding of complicated coupled nonlinear mechanics of flexible or soft structures with covered smart piezoelectric materials.