Corotational Unsymmetric Membrane Element Formulation for Geometric Nonlinear Analysis of Flexoelectric Solids Within the Consistent Couple Stress Theory

Abstract

In this work, the corotational (CR) penalty membrane element formulation for geometric nonlinear analysis of flexoelectric solids with large displacement but small deformation is proposed. To achieve this, the two-dimensional couple stress-based flexoelectric model is first extended to the CR form where the effect of rigid body rotation is excluded and, accordingly, the curvature that governs the electromechanical behavior of flexoelectricity is defined as the derivatives of the elastic rotation in the CR configuration. Next, a quadrilateral 8-node element is constructed based on the CR method. In the development, with the use of the penalty function method, the independent nodal rotation degrees of freedom are utilized to approximate the elastic rotation field, ensuring the higher-order continuity requirement for the displacement test function in a weak sense. In addition, to fully improve the element performance, the trial function of CR force-stress is formulated based on the polynomial force-stress functions, which are continually updated with reference to the latest deformed configuration. As demonstrated by the numerical benchmark examples, the new element can efficiently and accurately predict the geometric nonlinear electromechanical response of slender flexoelectric structures and capture the size effect very well.