Use of Piezoelectric Shear Response in Adaptive Sandwich Shells of Revolution - Part 2: Finite Element Implementation

A theoretical model has been presented in Part 1 of this paper for the use of piezoelectric shear response in adaptive sandwich shells of revolution. This second part presents its corresponding finite element implementation. For this, an isoparametric finite element discretization is retained for the cylindrical coordinates and independent mechanical and electric variables. These are interpolated with linear Lagrange polynomials along the meridian, but expanded with Fourier series along the shell circumference. This leads to a simple two-nodes conical finite element with 22 degrees of freedom. Straight meridian has been assumed to reduce eventual membrane locking, whereas the selective reduced integration has been used to avoid eventual shear locking. Therefore, prior to the element assembly, a geometric transformation from local curvilinear to global cylindrical coordinate system was made to take into account the shell meridian curvature and to handle its eventual discontinuities. After its validation, the present finite element has been used for vibration analysis of adaptive sandwich cylindrical shells and circular plates with open-and short-circuited embedded piezoelectric shear actuators. It was found that, contrary to extension actuators, shear ones are insensitive to the electric boundary conditions for axisymmetric modes. For non-axisymmetric ones, it was also found that bending modes dependence varies more than other radial, torsion and extension ones.