Effect of Porosity on the Modal Response of Doubly-Curved Laminated Shell Structures Made of Functionally Graded Materials Employing Higher Order Theories

The manuscript investigates the modal response of laminated anisotropic doubly-curved shell structures of variable thickness made of Functionally Graded Materials (FGM), according to an efficient equivalent single layer strategy where the displacement field is described employing a condensed unified formulation, accounting for a higher order through-the-thickness expansion. A generalized power law distribution is adopted for the assessment of the FGM layers, whereas the presence of voids is considered starting from different homogenization assumptions. Unlike previous works which presented a variation of the homogenized material properties within the shell solid, in this paper the problem of material porosity is addressed, therefore a general distribution of the volume fraction of the constituent materials is considered, along with the presence of voids. To this end, both linear and trigonometric through-the-thickness distributions are here assumed. The fundamental governing equations are derived starting from a proper set of curvilinear principal coordinates, while a generalized set of blending functions accounts for arbitrarily shaped structures. Non-conventional boundary conditions are, here, modelled with a distribution of linear springs along the shell edges. The numerical implementation of the problem is performed with the generalized differential quadrature method. A systematic set of validating examples is presented, whose results are compared to predictions based on refined models, as well as some experimental evidence. After a validation step, an extensive parametric analysis points out the sensitivity of the porosity parameters on the modal response of structures with different curvatures and external constraints, with valid findings for engineering design purposes.