Physical Mesomechanics. Vol. 27, No. 3, Pages 338-354, 2023

A. A. Daikh^{a, b}^,*^,**, M.-O. Belarbi^{c, d}, P. V. Vinh^e, M. Ladmek^b, A. Belkacem^b, M. S. A. Houari^b, H. M. Ahmed^f, and M. A. Eltaher^{g, h}

^aArtificial Intelligence Laboratory for Mechanical and Civil Structures, and Soil, University Center of Naama, P.O. Box 66, Naama, 45000 Algeria

^bLaboratory for the Study of Structures and Mechanics of Materials, Mustapha Stambouli University, B.P. 305, Mascara, 29000 Algeria

^cResearch Laboratory of Civil Engineering, University of Biskra, B.P. 145, Biskra, 07000 Algeria

^dDepartment of Civil Engineering, Lebanese American University, Byblos, Lebanon

^eDepartment of Solid Mechanics, Le Quy Don Technical University, Hanoi, Vietnam

^fDepartment of Civil and Environmental Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589 Saudi Arabia

^gFaculty of Engineering, Mechanical Design and Production Department, Zagazig University, Zagazig, 44519 Egypt

^hFaculty of Engineering, Mechanical Engineering Department, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589 Saudi Arabia

Abstract— This paper introduces a new shear deformation theory, employing the hyperbolic sine function, for exploring the free vibration properties of a novel functionally graded (FG) shell structure. The proposed theory ensures a parabolic distribution of shear strains and stresses across the thickness, with zero values at the top and bottom surfaces, eliminating the requirement for any shear correction factor. This is the first time such an approach has been utilized for studying this type of FG structure. The material properties are assumed to vary gradually across the thickness in the form of a trigonometric function. The proposed FG material stands out due to its excellent rigidity and smooth and continuous variation of the material components through the thickness. This composition has the potential to compensate for the deficiencies found in conventional FG sandwiches. Two types of functionally graded shells are considered: the trigonometric FG-A shell and the trigonometric FG-B shell. The governing equilibrium equations of the FG shell are derived in detail with the principle of virtual work and are solved analytically by the Galerkin method that can cover different boundary conditions. The proposed solution is constrained to rectangular and straight FG plates of uniform cross-section. A wide range of comparative studies is carried out to establish the accuracy and the performance of the present analytical model. A detailed parametric analysis is performed to highlight the influence of the material inhomogeneity parameter, geometry and various boundary conditions on the vibration response. The proposed model has an important role in the design of various vessels and shells.

Keywords: free vibration, FGM cosine shell, Galerkin technique, hyperbolic shear theory, analytical solutions

An Assessment of a New Hyperbolic Shear Deformation Theory for the Free Vibration Analysis of Cosine Functionally Graded Doubly Curved Shells under Various Boundary Conditions