# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 1 Introduction
studied sandwich beams embedding FGM in two ways: they applied the Ritz method and the principle of minimum total potential energy within the framework of [PERSON] and [PERSON] beam theories in [10] to assess the bending of beams with different cross sections; they also developed a four-node isoparametric beam element to study porous beams with FGM [11]. Concerning two-dimensional geometries, the same authors proposed the nonlinear analysis of FG shells in [12;13] by improving an isoparametric six-node TRIA element with strain interpolation functions; the mechanical properties grading followed a power low. They also proposed a three-node TRIA element [14] using a mixed strain finite element approach and demonstrated that it is possible to get the exact response of the beams with a low number of elements under large deformations. The solutions available in the literature seem to be promising. However, the results of a hygrometric stress analysis depend not only on the capabilities of the elastic model implemented but also on how the moisture content field has been evaluated, given the external boundary conditions. The hygrometric field acts as a field load; its quantification necessarily influences the stress analysis results. Aerospace applications usually involve thin components. Consequently, evaluating the moisture content field translates into determining its profile along with the thickness direction, generally coinciding with the grading direction of the mechanical/hygrometric properties. Developing a mechanical/elastic model for a thin component can be done at different levels of detail: 3D or 2D approaches, coupled with an analytical or numerical method. However, this might not be sufficient in defining the refinement of a model when hygrometric stresses are concerned. Even an analytical, exact 3D model would give wrong results when fed from an inaccurate moisture content field. The molecular diffusion depends on the gradient of the concentration and is described by [PERSON]'s law. A solid mathematical analogy between [PERSON]'s law and the Fourier heat conduction equation strongly simplifies the analysis and the dissertation. An exact solution comes from resolving three-dimensionally the diffusion/conduction equations; however, simplified solutions might benefit from their unidirectional simplification or an assumed-linear profile. Those three situations require an external evaluation of the moisture content field before the mechanical analysis. An alternative consists in defining a coupled hygro-mechanical model, in which the moisture content field is a primary variable of the problem in analogy with the thermal field [15; 16; 17; 18; 19; 20]. This paper discusses an hygro-elastic shell model, which relies on the exponential matrix method. This model does not limit to a specific geometry or lamination scheme; on the contrary, it handles plates, cylindrical, and spherical shells. Furthermore, it accepts one or more FGM layers on their own or coupled with homogeneous layers (sandwiches give an example). It extends the authors' 3D exact thermo-elastic shell model discussed in [21] to hygro-elastic stress analysis. The problem is defined under steady-state conditions only; the solution requires the moisture content amplitudes at the top and bottom external surfaces to be specified. As a first step, the solution evaluates the moisture content profile through the thickness direction. The authors considered three possible options: the 3D Fick diffusion law, its 1D simplified version, and an a priori assumed linear trend. Despite this, the model handles the solution via a layer-wise approach and through the exponential matrix method. The present authors are not the only ones using the exponential matrix method for solving the differential equilibrium equations. [PERSON] and [PERSON] [22] already used it in analyzing the free vibrations of cylinders; [PERSON] [23] applied it to study multilayered plates.