# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 1 Introduction
The author deepened the influence of moisture content field and temperature on the best-buckling response of beams. A nonlinear finite element solution was considered, in which the authors handled the kinematics of the best-buckling through the Lagrangian approach. More substantial is the literature discussing the moisture content effects on isotropic, orthotropic, and laminated structures. [PERSON] and [PERSON] [43] studied multilayered plates and proposed a two-dimensional analytical solution for hygrothermal effects on multilayered plates. The Classical Lamination Plate Theory and an analytical 3D plate model were compared in [44; 45] while performing the hygro-thermal stress analysis. The CLT also received the attention of [PERSON] [46] when he investigated composite plates. [PERSON] [47] made an analytical investigation of composite cylindrical shells, while [PERSON] [48] focused on buckling and post-buckling behaviors. The CLT was found to be inadequate in hygrothermal mechanical analysis by [PERSON] and his colleagues [49]. There is no shortage of numerical models in this field; an example is given in [50], where plates with a central hole were studied. Multilayered structures under hygrometric fields have been the target of the finite element model by [PERSON] et al. [51]; [PERSON] and [PERSON] studied buckling and vibration in multilayered shells with double curvature [52] through a finite element model grasping the hygrothermal effects. They extended this research to dynamic instability of doubly-curved shells embedding composite materials through a nonlinear finite element under orthogonal curvilinear coordinates [53]. [PERSON] and [PERSON] included the Fick diffusion law [54] into a FE shell model devoted to isotropic and multilayered structures under a hygrothermal environment. With the same aim, [PERSON] and [PERSON] [55] investigated cylindrical shell panels under large deflections through a higher-order shear deformation theory. [PERSON] also proposed a higher-order FE for laminated parts [56]. [PERSON] et al. [57; 58; 59] considered the elastic properties variation in composite plates with the temperature and the moisture content and discussed a transient hygroscopic stress analysis. In [60; 61; 62; 63], simplified solutions for hygro-thermal stresses analysis of composite plates considering the variation in elastic properties due to the hygrothermal environment are also considered. An analogous evaluation devoted to dynamic behavior has been proposed in [64; 65]. [PERSON] [66] used a FE model to investigate how a severe hygrothermal environment can affect the initiation and progress of damages in composites. The literature survey highlighted that the hygromechanical stress analysis of structures embedding FGM layers still misses general and exact solutions as benchmarks in new solutions. Instead, the only analytical models work on defined and specific boundary conditions, laminations, and geometry. This paper intends to fill this gap by extending the authors' previous work on multilayered structures to FGMs. The manuscript is organized as follows. Section 3 describes the hygro-elastic shell problem and its solution with the exponential matrix method. Section 2 explores the moisture diffusion problem with the three approaches introduced before. The Fick law of diffusion has the same mathematical formulation of the Fourier heat conduction equation; the moisture diffusion problem and the heat conduction problem are in analogy, indeed, as demonstrated in [67], and further confirmed by [PERSON] and [PERSON] [68; 69]. Therefore, Section 2 quantifies the moisture content profiles. Solving the 3D Fick diffusion law is possible by exploiting the analogy with the heat conduction problem. [PERSON] and [PERSON] [35] give a methodology that can also be applied in this context; the unidimensional Fick diffusion law disregards the diffusion fluxes in directions other than the thickness coordinate and can be calculated in analogy. Section 4 gives two sets of results.