Abstract
This paper presents a model called SAM-FG (stress approach model of functionally graded shells) for linear elastic, thin, and moderately thick shells made of functionally graded materials. The model is an extension of the SAM-H model, originally created for homogeneous shells. Assuming that the material is orthotropic and that one of its orthotropic directions is the thickness direction, the extension consists in considering that the 3D compliance tensor may depend on the thickness coordinate. The model starts with a tunable polynomial approximation of the 3D stress field that contains the same generalized forces as SAM-H. This stress approximation verifies the 3D equilibrium equations and the stress boundary conditions at the faces of the shell. As in SAM-H, 5 generalized displacements appear in SAM-FG. By applying the Hellinger–Reissner functional and Reissner’s variational method, the generalized forces, strains, and equations in SAM-FG turn out to be the same as in SAM-H, except for the generalized constitutive equations. To prove the accuracy of the model, SAM-FG is first applied to a simply supported, functionally graded plate and its results are compared to other models. To validate the model for shell-like structures, SAM-FG results are compared to those obtained with solid finite element calculations for three case studies of structures subjected to an internal pressure. The first one deals with a hollow sphere made of an isotropic functionally graded material. The second case considers a hollow cylinder made of an orthotropic functionally graded material. In the last case, a catenoid with an isotropic functionally graded material is studied. In all cases, the mean displacements are correctly predicted, even if the main purpose of the SAM-FG model is not to calculate these fields accurately. The stress field approximations are very accurate, and since the implementation of the shell model in a finite element code would imply 5 degrees of freedom per node, SAM-FG is a good alternative to solid finite element calculations for the structural analysis of functionally graded shells with a reasonable computational cost.
Highlights
Academic Editor: Marek Lefik is paper presents a model called SAM-FG for linear elastic, thin, and moderately thick shells made of functionally graded materials. e model is an extension of the SAM-H model, originally created for homogeneous shells
A new model called SAM-FG was developed for elastostatic problems of functionally graded shells whose thickness may range from thin to moderately thick. is model is inspired by the SAM-H model conceived by Domınguez-Alvarado and Dıaz-Dıaz in [34] and limited to homogeneous shells
In contrast to the stress approximation in SAM-H, the degree of the polynomial expansion used in SAM-FG to approximate the stress field across the shell thickness is variable. is degree is given by a polynomial fit of order r of the components of the in-plane stiffness matrix. e polynomial degrees of the approximation of in-plane stresses, out-ofplane shear stresses, and out-of-plane normal stresses are r + 1, r + 3, and r + 4, respectively. e stress field verifies the boundary conditions at the faces of the shell and the 3D equilibrium equations. e use of Hellinger–Reissner functional allowed to identify 5 generalized displacements that are identical to those in SAM-H. e application of Reissner’s variational theorem yielded the generalized equilibrium equations, boundary conditions, and constitutive equations
Summary
Carlos Humberto Rubio Rascon, Alberto Dıaz Dıaz ,1 and Axel Fernando Domınguez Alvarado. Nguyen et al [17] proposed a model based on the first-order shear deformation theory (FSDT) applied to functionally graded material plates using shear correction factors (SCFs) to calculate accurately shear forces and strains. Other models equivalent to the FSDT are those developed by Zenkour [19] and Touratier [20] that include 5 generalized displacements and do not need SCF Another interesting model was developed by SiddaRedddy et al [21]; it is based on a higher-order shear deformation theory (HSDT) using 9 degrees of freedom and proved to yield accurate results for FGM plates. In [31], Kulikov and Plotnikova applied his SaS method to the study of static shells and validated the method for 7 reference surfaces by comparing the model results with those of SFE in the case of a homogeneous cylinder. The numerical results of SAM-FG are compared to those of SFE results in order to test the accuracy of the model
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