## 1 Introduction  
The development of refined structural models or higher-order theories currently represents one of the most active areas for many researchers in the structural mechanics field. Their interest in this subject is mainly due to the fact that the use of advanced materials has shown inadequacy of the classical or first-order theories to capture the effective mechanical behavior of this class of materials, such as laminated composites or functionally graded materials [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. In these particular circumstances, the fundamental hypotheses of a lower-order model could lead to inaccurate results since some effects are neglected or not even considered. A typical situation is the case of structures made of a combination of highly heterogeneous materials in the lamination scheme. As highlighted in the work by [PERSON] and [PERSON] [21], many refined approaches arose to predict a more coherent structural behavior. For the sake of completeness, the readers can find some examples of these approaches in [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. It should be noted that the aforementioned papers were mainly focused on plates and shells. The insufficiency of classical plate and shell theories can also be highlighted in the dynamic analysis. In order to evaluate the natural frequencies of composite structures correctly,several higher-order shear deformation theories (HSDTs) were developed. In general, two different approaches are followed, namely, equivalent single layer (ESL) and layer-wise (LW) models (see [PERSON] [1] for a brief review). The former is able to study a composite structure by referring each geometric and mechanical parameter on the middle surface of the structure, which is taken as reference domain for the governing equations. In a similar manner, the degrees of freedom of the problem under consideration are evaluated on the middle surface, independently from the enrichment of the kinematic model [33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46]. On the other hand, the latter approach takes into account each layer (or ply) that composes the structure separately from the others. In general, a LW model defines the kinematic expansion, and consequently the degrees of freedom, along the thickness of each layer. As a result, this approach is able to deal with continuous displacements but characterized by discontinuous derivatives at the interfaces [46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. The works [57; 58; 59] represent a turning point in the advancement of HSDTs. In fact, they provided the basis for the development of a unified formulation (UF), which is able to analyze and use several structural models within a single framework. As a matter of fact, the order of kinematic expansion can be arbitrarily chosen in order to obtain the desired theory that is more appropriate to deal with a particular physical phenomenon. In the review paper [57] a complete treatise about both ESL and LW approaches is presented. It is important to cite also the works [58; 59], where the so-called zig-zag theories for layered structures were discussed. For the sake of completeness, it should be recalled that a zig-zag theory is actually an ESL model embedded with a specific function that allows to consider continuous through-the-thickness displacements characterized by different slope at the interface between two plies, similar to a LW theory [31; 32; 33]. This aspect is commonly denoted as zig-zag effect and it is generally given by the choice of different materials along the transverse direction of the structure. For instance, this effect is particularly evident when a soft-core is included into the lamination scheme of a generic plate or shell (sandwich structure). In general, the Murakami's function is employed for this purpose. Recently, a huge number of scientific papers based on the UF is available in the literature.