3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles

Abstract

The exact three-dimensional (3D) shell model proposed in the present paper is able to perform the thermal stress analysis of simply-supported Functionally Graded Material (FGM) spherical and cylindrical shells, cylinders and plates. The model is based on the 3D equilibrium equations for spherical shells developed using an orthogonal mixed curvilinear coordinate system. The use of this reference system allows the investigation of cylindrical shells, cylinders and plates as particular cases of spherical shells by means of simple considerations on the radii of curvature. The 3D shell model uses a layer-wise approach and the exponential matrix method to calculate the general and the particular solutions through the thickness direction z. The system of second order differential equations in z is not homogeneous because of the thermal terms which are externally defined. The system is reduced to a group of first order differential equations in z simply redoubling the number of variables. The solution is in closed form in the in-plane directions α and β because of the hypotheses of simply-supported boundary conditions, harmonic forms for displacement and temperature fields, and isotropic behavior in the in-plane directions for functionally graded materials. In order to define the equivalent thermal load, the temperature profile through the thickness is separately defined by means of three possible ways. Using the hypothesis of temperature amplitudes imposed at the top and bottom external surfaces in steady-state conditions, the temperature profile can be: imposed as linear through the entire thickness direction, calculated by solving the 1D version of the Fourier heat conduction equation, or calculated by solving the 3D version of the Fourier heat conduction equation. The effects of different temperature profiles on the displacement and stress analyses of FGM plates and shells are here remarked. The first order differential equation system in z has not constant coefficients because of the presence of radii of curvature for shells and through-the-thickness variable elastic and thermal coefficients for the FGM layers. An appropriate number of mathematical layers is introduced to calculate the curvature influence for shells and the elastic and thermal material coefficients for FGM layers. Therefore, the system can be considered as differential equations with constant coefficients. The proposed results allow the evaluation of thickness ratio, geometry, lamination scheme, thickness material law and temperature profile effects in the related thermal stress analysis of single-layered and sandwich FGM plates, cylinders, spherical and cylindrical shells.