# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 1 Introduction
The orthogonal mixed curvilinear coordinate system helped study spherical shells in [24] using three transverse stress and three displacement components as primary variables of the problem. The analytical procedure is in analogy with the free vibration analysis and the static analysis under mechanical load discussed by the first author in [25; 26; 27; 28; 29; 30]. Those previous formulations already handled different materials and geometries but lacked loads other than the mechanical ones. In those simpler cases, a set of three homogeneous differential equations are at the basis of the problem. However, the hygrometric load adds an additional term that makes them not homogeneous. The exponential matrix method also handles this feature, as discussed in [21] for the thermoelastic stress analysis of shells with FGMs. The closed-form solution of the problem is possible given the harmonic forms imposed on the variables (displacements and hygrometric field) and the simply supported boundary conditions. Moreover, the whole formulation benefits an orthogonal mixed curvilinear coordinate system, following the suggestions in [31; 32; 33; 34]. This strategy introduces a set of curvature terms, the elements through which the equations automatically adapt to the different geometries. Such terms are a function of the thickness coordinate, together with the elastic and hygrometric coefficients in FGM layers. Introducing a set of fictitious (mathematical) layers allows obtaining constant coefficients and using the method discussed in [35] to solve the problem. The literature overview offers different analytical and numerical 2D models handling the hygrometric stress analysis; they specifically focus on multilayered structures. Far fewer discuss the problem of structures embedding FGMs. The literature overview will demonstrate that, to the authors' knowledge, no analytical 3D shell model exists, in which the structures are different, provided that they have constant radii of curvature, and the moisture content evaluation follows three different approaches. [PERSON] [36] studied rectangular plates embedding FGMs when subjected to different boundary conditions, including moisture content and temperature field. They developed an analytical method through the hyperbolic shear deformation plate model, which satisfied the stress boundary conditions and required no shear corrections. The volume fractions of the ceramic-metal constituents were used to define the materials grading in the thickness direction following a power law. The same power law was included in [PERSON]'s work [37], devoted to studying how the hygothermal environment affects the vibration characteristics of plates embedding FGMs. To do this, the author developed a finite element model of the structures under investigation. [PERSON] [38] and his colleagues focused on circular plates with a variable thickness along with the radial direction. They derived nonlinear governing equations for temperature, moisture content, and displacement fields and solved them through the differential quadrature method. [PERSON] also studied how the thermal and hygrometric fields affect the bending and the buckling of plates embedding FGMs [39]. In this case, also, the authors considered a material grading in the thickness direction of the structure. Their formulation relies on an exponential shear deformation theory and applies to plates resting on elastic foundations. [PERSON]'s principle was used to derive the equilibrium equations, and [PERSON]'s method to obtain the results. [PERSON] also studied plates resting on elastic foundations, embedding FGMs [40]. The author developed a novel higher-order shear deformation theory model, deducing the problem equations through the virtual work principle. Hygro, thermal, and mechanical loads are all properties that can be handled. Analogously, [PERSON] [41] developed a further model for the same issue. In this case, the author used a four-variable refined plate theory. Analogous research has been proposed in [42] concerning functionally graded beams.