A quadrilateral shallow shell element based on the third-order theory for functionally graded plates and shells and the inaccuracy of rule of mixtures

• A quadrilateral element developed for shallow FGM shells, using third order theory.
• Better accuracy than other 2D finite elements with the least computational cost.
• An assessment of accuracy of rule of mixtures (ROM) is made with Mori–Tanaka model.
• The ROM yields inaccurate response when constituents' properties are very different.
• The error in ROM predictions depends on geometry parameters and boundary conditions.

A four-node quadrilateral element is developed for the dynamic analysis of doubly curved functionally graded material (FGM) shallow shells, using the refined third order theory. Two micromechanics models, the Voigt's rule of mixtures (ROM) and the Mori–Tanaka model, are considered for computing the effective material properties at a point. The accuracy of the element is examined by comparing with various three dimensional elasticity and two dimensional (2D) analytical and finite element solutions available in the literature for static and free vibration responses of FGM plates and shells. It is shown that the present element, with the least number of degrees of freedom, achieves similar or better accuracy compared to other available 2D finite elements some of which are even based on higher order theories. Using this element, we also make a systematic assessment of the accuracy of the widely used ROM in predicting the behavior of FGM structures, for different values of the inhomogeneity parameter, and different geometrical parameters, boundary conditions, and material combinations. It is revealed that there can be very significant error in the deflection, stresses and natural frequencies predicted by the ROM, depending primarily on the inhomogeneity parameter and the difference in the material properties of the constituents.