An analytical method for free vibration analysis of multi-directional functionally graded porous doubly-curved shells in thermal environment with various boundary conditions

This paper uses the analytical method with arbitrary boundary conditions to model and analyze the free vibrations of multi-directionally functionally graded porous (MFGP) doubly-curved shallow shells resting on the Pasternak foundation in a temperature environment. It is anticipated that all mechanical parameters, except Poisson's ratio, will change in the direction of length, width, and thickness. To comprehensively describe the shell's displacement, strain, and stress fields, a modified first-order shear deformation theory (FSDT) with an assumption of cosine distribution shear stresses has been developed. The fact that the enhanced FSDT theory does not require the use of shear correction and that the shear stress at the two free faces of the shells is zero are two of the theory's most significant advantages. Using Hamilton's principle and improved FSDT, one may get the governing equation for free vibration analysis of MFGP doubly-curved shallow shells. The Galerkin approach is proposed to solve the governing equation of MFGP doubly-curved shallow shells with various boundary conditions. The trustworthiness of the article is evaluated via its publication in the article model's several special cases. From this point on, a collection of findings about the natural frequency of MFGP doubly-curved shallow shells is identified and shown in the form of tables and graphs. The results provided in this manuscript can be used as a benchmark solution for further studies as far as the vibration behavior of the MFGP doubly-curved shallow shells is concerned.