An isogeometric degenerated shell formulation for geometrically nonlinear analysis of smart laminated shells

The geometrically nonlinear solution of laminated and piezolaminated shells with the isogeometric method (IGA) is the subject of this paper. Usage of same models for modeling and analysis in IGA, and its advantages in exact representation of the geometry make it attractive for shell analysis. From different classes of basis functions used in IGA, NURBS are selected in this work. Despite the advantages of IGA over FEA, the definition of director vectors at the control points that are located outside the geometry is a challenging problem. In this work, the initial director vectors at control points are obtained by solving a defined system of equations at the Greville points that result in an excellent convergence rate for high-order NURBS. The components of director vectors at Greville points are considered as known values of equations. A nonlinear formulation is written based on the Reissner-Mindlin assumption and updated Lagrangian approach. The electric field is approximated using a sub-layer approach. The governing equation for the piezolaminated shell is derived using the Lagrangian conjugate tensors in terms of through the thickness parameter. Thus, an analytical integration can be performed in each layer which results in less computational cost. To evaluate the accuracy of the present formulation, several well-established linear and nonlinear benchmark problems incorporating various boundary conditions, geometries and material properties are considered.