Various Higher-order Shear Deformation Theories for Vibration Analysis of Functionally Graded Plates

In the present paper, the four various higher-order shear deformation theories are used to analyze the free vibration of functionally graded plate. A navier-type analytical method is used to solve the governing differential equations[1]. Natural frequencies of simply supported functionally graded plates are calculated. The present results are compared with the available published results which verify the accuracy of various higher-order theories. The influences of side to thickness ratio and power law index on the fundamental frequencies of a simply supported square functionally graded plate are also studied. Introduction The material properties of the fiber-reinforced laminated composite materials are discontinuous across adjoining layers which result in the delaminating mode of failure. Functionally graded plates can overcome the delaminating mode due to their continuous variation of material properties from one surface to another[2]. The functionally graded material for high-temperature applications may be composed of ceramic and metal. This paper uses various shear deformation theories of Touratier (1991), Mantari (2012), Karama (2003), Levinson (1980) to study the free vibration behavior of functionally graded plates. A navier-type analytical method is used to solve the governing differential equations. The present results are compared with those of Vel and Batra (2004) and Matsunaga (2008)[3,4]. Governing Equations and Boundary Conditions Displacement Field The displacement field of the higher order shear deformation theory is: ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) x y w x y U u x y z f z x y x w x y V v x y z f z x y y W w x y φ φ ∂ = − + ∂ ∂ = − + ∂ = (1) where , , , x u v w φ and y φ are the five unknown displacement functions of middle surface of the plate. h is the thickness of the plate. The Transverse Shear Function The transverse shear function in Touratier (1991) is: ( ) sin( ) h z f z h π π = (2) The transverse shear function in Mantari (2012) is: 405 1 cos( ) 2 ( ) sin( ) 2 z h z f z e z h h π π π = + (3) The transverse shear function in Karama (2003) is: 2 2( / ) ( ) z h f z ze− = (4) The transverse shear function in Levinson (1980) is: