This paper presents a general solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic and composite structures. Plates, cylinders, cylindrical and spherical shells are analysed using mixed orthogonal curvilinear coordinates and simply-supported boundary conditions. The 3D equilibrium equations, written for spherical shells, automatically degenerate in those for simpler geometries which can be seen as particular cases. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method and the mathematical layers to solve the equilibrium differential equations through the thickness direction. The analysed shell structures are divided into several mathematical layers in order to evaluate the curvature terms of the shell geometry. The closed-form solution is obtained assuming harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady state conditions. Then, the temperature profile is evaluated through the thickness direction. Three different methods are here employed to define the temperature profile through the thickness: - calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; - evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; - a priori assumed linear temperature profile through the thickness direction ranging from the bottom temperature value to the top temperature value. The 3D calculated temperature profile allows to consider both thickness layer and material layer effects, while the 1D calculated temperature profile is only able to evaluate the material layer effect. The linear assumption for the temperature profile is only valid for thin and one-layered structures. Once the temperature profile is defined at each thickness coordinate, it is considered as a constant and known term in the differential equilibrium equations written for each mathematical layer. The obtained system consists in a set of second order non-homogeneous differential equilibrium equations. After a reduction to the first order, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and their first derivatives made with respect to the thickness coordinate. Then, the strain and stress components are calculated using the strain-displacement relations for shells with constant radii of curvature and the linear thermo-mechanical constitutive equations, respectively. The solution is validated using reference results available in the literature. Several analyses, in terms of displacements, in-plane and out-of-plane stresses, strains, temperature profiles and heat fluxes, are presented for different thickness ratios, geometries, lamination schemes and materials. Results show the importance of both the correct definition of the elastic part in the 3D shell model and the correct evaluation of the temperature profile through the thickness of the structure. Even if a very refined 3D shell model is employed, wrong thermo-mechanical responses can be obtained if the temperature profile through the thickness is not coherently defined.

Thermo-mechanical analysis of composite structures via a 3D exact shell model / Brischetto, Salvatore; Torre, Roberto. - (2018). (Intervento presentato al convegno ICCS 21: 21st International Conference on Composite Structures tenutosi a Bologna (Italy) nel 4-7 September 2018).

### Thermo-mechanical analysis of composite structures via a 3D exact shell model

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*Salvatore Brischetto;Roberto Torre*

##### 2018

#### Abstract

This paper presents a general solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic and composite structures. Plates, cylinders, cylindrical and spherical shells are analysed using mixed orthogonal curvilinear coordinates and simply-supported boundary conditions. The 3D equilibrium equations, written for spherical shells, automatically degenerate in those for simpler geometries which can be seen as particular cases. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method and the mathematical layers to solve the equilibrium differential equations through the thickness direction. The analysed shell structures are divided into several mathematical layers in order to evaluate the curvature terms of the shell geometry. The closed-form solution is obtained assuming harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady state conditions. Then, the temperature profile is evaluated through the thickness direction. Three different methods are here employed to define the temperature profile through the thickness: - calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; - evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; - a priori assumed linear temperature profile through the thickness direction ranging from the bottom temperature value to the top temperature value. The 3D calculated temperature profile allows to consider both thickness layer and material layer effects, while the 1D calculated temperature profile is only able to evaluate the material layer effect. The linear assumption for the temperature profile is only valid for thin and one-layered structures. Once the temperature profile is defined at each thickness coordinate, it is considered as a constant and known term in the differential equilibrium equations written for each mathematical layer. The obtained system consists in a set of second order non-homogeneous differential equilibrium equations. After a reduction to the first order, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and their first derivatives made with respect to the thickness coordinate. Then, the strain and stress components are calculated using the strain-displacement relations for shells with constant radii of curvature and the linear thermo-mechanical constitutive equations, respectively. The solution is validated using reference results available in the literature. Several analyses, in terms of displacements, in-plane and out-of-plane stresses, strains, temperature profiles and heat fluxes, are presented for different thickness ratios, geometries, lamination schemes and materials. Results show the importance of both the correct definition of the elastic part in the 3D shell model and the correct evaluation of the temperature profile through the thickness of the structure. Even if a very refined 3D shell model is employed, wrong thermo-mechanical responses can be obtained if the temperature profile through the thickness is not coherently defined.##### Pubblicazioni consigliate

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`https://hdl.handle.net/11583/2712410`