3D thermal stress analysis in functionally graded plates and shells

The proposed work investigates the thermal stress analysis of one-layered and multi-layered Functionally Graded Material (FGM) plates, cylinders, cylindrical shells and spherical shells. FGMs here considered have variable through-the-thickness thermal and elastic properties. A threedimensional (3D) exact solution is proposed for simply-supported sides and harmonic forms for displacements, stresses and temperature. The employed 3D shell model is based on a set of 3D equilibrium equations for spherical shells written in mixed orthogonal curvilinear coordinates. These equations degenerate in those for simpler geometries, such as plates and cylinders, by means of general considerations about the radii of curvature. FGM plates and shells are analysed in terms of displacements, stresses, strains, temperature profile and heat fluxes when temperature amplitudes are imposed at the external surfaces in steady state conditions. Consequently, three different methods are here employed to evaluate the temperature profile through the thickness direction. Firstly, the 3D version of Fourier heat conduction equation is analytically solved considering the effects of the in-plane dimensions and thickness dimension of the structure. Secondly, the 1D version of the Fourier heat conduction equation is solved using the continuity conditions for temperature and transverse normal heat flux at each layer interface without considering in-plane dimensions of the structure. The third possibility is the “a priori” assumption of a linear temperature profile through the thickness. The 3D version of Fourier heat conduction equation allows the evaluation of both thickness and material layer effects. The 1D version of Fourier heat conduction equation is able to consider only the material layer effect. On the contrary, the “a priori” assumption of a linear temperature profile is valid only for thin and one-layered homogeneous structures. After the opportune definition of the temperature profile through the thickness direction, this field load becomes a constant and known term in the proposed second order non-homogeneous differential equations. The obtained system is reduced to a system of first order differential equations in the thickness coordinate z simply doubling the number of variables (displacements and relative derivatives made with respect to the thickness coordinate z). The non-homogeneous equations are solved using the exponential matrix method for both the general and the particular solution. The exponential matrix is opportunely calculated choosing the appropriate value for the N order of expansion. An opportune M number of mathematical layers is introduced to solve 1D and 3D versions of Fourier heat conduction equation and the system of non-homogeneous first order partial differential equations. In these cases, equation coefficients are not constant and M mathematical layers are mandatory to opportunely evaluate curvature terms of shell geometries and the through-the-thickness FGM law for elastic and thermal properties. Obtained results will demonstrate the importance of the correct evaluation of the temperature profile through the thickness of one-layered and multi-layered FGM structures in order to have a satisfactory 3D thermal stress analysis. In fact, a refined elastic part in the 3D shell model, used for the appropriate thermal stress analysis of FGM plates and shells, could be not sufficient if the temperature profile is not coherently defined with the lamination, thickness ratio and FGM law of the analysed structure.