Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

Abstract

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé's coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.

A quasi-3D hyperbolic shear deformation theory for functionally graded plates

“…The present quasi-3D theory is even more accurate than the quasi-3D sinusoidal theory [33]. Since the present quasi-3D theory and other quasi-3D theories include the thickness stretching effect, their solutions are very close to each other.…”

“…The obtained results are compared with the exact 3D [33] and quasi-3D solutions [28,[33][34]. It should be noted that the quasi-3D solutions [33][34] are derived based on a trigonometric variation of both in-plane and transverse displacements, while the quasi-3D solutions [28] are computed based on a cubic variation of in-plane displacements and a parabolic variation of transverse displacement across the thickness. In addition, the results of HSDT [35] are also provided to show the importance of including the thickness stretching effect.…”

“…In general, the present quasi-3D theory is highly accurate and comparable to 3D solution even in the case of very thick plates, e.g., / 2 a h = . It is worth noting that the developed theory consists of five unknowns, while the number of unknowns in the HSDT [35] and other quasi-3D theories [28,[33][34] is five and six, respectively. Consequently, it may be concluded that the present quasi-3D theory is not only more accurate than the HSDT having the same five unknowns, but also comparable with the quasi-3D theories having more number of unknowns.…”

“…The effective Young's modulus ( ) E z is assumed to vary exponentially through the thickness of the plate as [33] ( ) ( ) ( ) …”

A quasi-3D theory for vibration and buckling of functionally graded sandwich beams

“…In order to include shear deformation and thickness stretching effects, the quasi-3D theories, which are based on a higher-order variation through the thickness of the in-plane and transverse displacements, are used. By using these theories, although a lot of work has been done for isotropic and sandwich FG plates ( [5]- [14]), the research on FG sandwich beams is limited. Carrera et al [15] developed Carrera Unified Formulation (CUF) using various refined beam theories (polynomial, trigonometric, exponential and zig-zag), in which non-classical effects including the stretching effect were automatically taken into account.…”