Free and Forced Vibration of Laminated and Sandwich Plates by Zig-Zag Theories Differently Accounting for Transverse Shear and Normal Deformability

A number of mixed and displacement-based zig-zag theories are derived from the zig-zag adaptive theory (ZZA). As a consequence of their different assumptions on displacement, strain, and stress fields, and layerwise functions, these theories account for the transverse shear and normal deformability in different ways, but their unknowns are independent of the number of layers. Some have features that are reminiscent of ones that have been published in the literature for the sake of comparison. Benchmarks with different length-to-thickness ratios, lay-ups, material properties, and simply supported or clamped edges are studied with the intended aim of contributing toward better understanding the influence of transverse anisotropy on free vibration and the response of blastloaded, multilayered, and sandwich plates, as well as enhancing the existing database. The results show that only theories whose layerwise contributions identically satisfy interfacial stress constrains and whose displacement fields are redefined for each layer provide results that are in agreement with elasticity solutions and three-dimensional (3D) finite element analysis (FEA) (mixed solid elements with displacements and out-of-plane stresses as nodal degrees of freedom (d.o.f.)) with a low expansion order of polynomials in the in-plane and out-of-plane directions. The choice of their layerwise functions is shown to be immaterial, while theories with fixed kinematics are shown to be strongly case-sensitive and often inadequate (even for slender components).