Higher order theories for the modal analysis of anisotropic doubly-curved shells with a three-dimensional variation of the material properties

The manuscript investigates the dynamic properties of doubly-curved shell structures laminated with innovative materials employing the Generalized Differential Quadrature (GDQ) method. A higher order generalized expression of the displacement field variable is assumed following the Equivalent Single Layer (ESL) approach. The geometrical description of the structures follows the differential geometry basics, whereas arbitrarily-shaped domains are distorted by means of generalized isogeometric blending functions. A three-dimensional generally anisotropic elastic constitutive behaviour is assumed within each layer of laminates, and a smooth variation of the material properties is implemented. In particular, a two-dimensional distribution alongside the physical domain is considered, whereas a through-the-thickness dispersion of the material properties is associated to each layer, taking into account both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian principle, and they are solved directly in strong form via the GDQ method taking into account a non-uniform discrete computational grid. After some preliminary validating examples, some parametric investigations are performed systematically, showing the influence of the variation of the material properties on the modal response of the structures characterized by different lamination schemes, curvatures and boundary conditions. The proposed model allows for the dynamic analysis of the structures of different curvatures characterized by very complicated dispersions of the material properties following a simple and accurate methodology.