Natural Frequencies of Shear Deformable Porous Orthotropic Laminated Doubly-Curved Shallow Shells with Non-Uniformly Distributed Porosity Using Higher-Order Shear Deformation Theory

Abstract

Porous materials, characterized by a network of voids, offer unique functional properties such as reduced density, high stiffness-to-weight and strength-to-weight ratios, and resistance to mechanical and thermal shocks. Despite their advantages, there is a lack of study on the free vibration behavior of porous laminated doubly-curved shallow shells. This study investigates the natural frequencies of porous orthotropic laminated doubly-curved shallow shells with different porosity distribution patterns. Porosity-dependent material properties are graded in the thickness direction using special cosine and sine functions. The equations of motion are established using Hamilton's principle based on the higher-order shear deformation theory and solved by Galerkin's method and an auxiliary function of simply supported boundary conditions to achieve natural frequency formulation. The formulation is confirmed by comparing the results with those in the existing literature, and an excellent agreement is observed. A parametric study is employed to analyze the effect of porosity coefficients, porosity distribution patterns, orthotropy, lamination sequences and orientations, shallow shell types, and geometrical characteristics on the natural frequencies of porous orthotropic laminated doubly-curved shallow shells. The results reveal that porosity distribution significantly influences dynamic behavior. Uniform porosity distribution (UDP) exhibits consistent effects across parameters, while non-uniform distributions (NUDP1, NUDP2, NUDP3) induce complex, often wavy variation, particularly in hyperbolic paraboloidal shells (HPS). Spherical shells (SS) consistently show higher natural fundamental frequencies (NFNFs) compared to HPS, with this difference becoming more pronounced with increasing aspect ratios and diminishing orthotropy ratios. Unexpectedly, single-layered configurations in HPS sometimes exhibit higher N-FNFs than multi-layered ones, a contrast to the behavior observed in SS. Orientation angles also play a critical role, with certain ranges leading to wavy variations in N-FNFs, particularly under non-uniform porosity distributions. These findings highlight the intricate interplay between porosity, lamination, geometry, and orthotropy, offering valuable insights for optimizing porous material design in advanced engineering structures.