Modeling And Mechanical Characteristics Of Functionally Graded Plates Based On High-order Shear Deformation Theory

The use of advanced composite materials is an inevitable step to achieve lightweighting in the ship industry.Functionally graded materials（FGM）have great potential applications in ship and ocean engineering due to the smooth and continuous distribution of material properties in one or more directions.These features help to overcome the problems of delamination and stress concentration between layers of traditional fiber-reinforced composites.Therefore,it is a very important to use analytical modelling to study the static and dynamic responses,such as the bending,vibration and buckling characteristics of FGM plates with accuracy and systematically study the effect of material parameters and geometric parameters on mechanical characteristics.In this paper,the modelling of FGM plates is studied based on the higher-order shear deformation theory,a unified model of higher order shear deformation theory is also established,and the static and dynamic characteristics of FGM plates are given.The main results of the research are as follows:To address the problem that the bending response of functional graded plates cannot be predicted accurately based on shear deformation theories,the influence of the gradient index on the distribution of transverse shear strain is considered,and the concept of shape parameter m is introduced into the transverse shear function and the optimal value of this parameter is assumed to vary with the power-law indexes,making the distribution of transverse shear strain in the thickness direction of functionally graded plates more accurate.The value of the optimal shape parameter m_opis determined by the golden section search.This shear strain shape function satisfies the stress-free boundary condition on the top and bottom surfaces of the FGM plates without using any shear correction factors.The differential governing equations and boundary conditions are derived from Hamilton’s principle to obtain a Navier-type closed-form solution for simply-supported boundary conditions.The accuracy of the proposed theory is verified by comparing the obtained results with the existing results.In addition,the effect of gradient index,side-to-thickness ratio and aspect ratio on the static responses is also studied.A novel simplified modeling method is proposed for the free vibration analysis of functionally graded plates.The effect of the gradient indexes on the transverse shear strain distribution is considered,two new transverse shear functions containing a shape factor m are given for dynamic modelling of FGM plates.The transverse shear functions satisfy the stress-free boundary conditions at the top and bottom surfaces of the FGM plates,so that the use of transverse shear correction factors is not required.The material properties of the FGM plates vary continuously according to the power-law distribution and Mori-Tanaka distribution through the thickness of the FGM plates.The displacement field is expressed as undetermined integral terms and this can improve the computational efficiency.The differential governing equations and boundary conditions are obtained based on Hamilton’s principle.The dynamic problem is transformed into a solution problem of the eigenvalue equation based on the Navier solution.The obtained obtained results are compared with other results to verify the accuracy and efficiency of the present theory.The effect of thickness stretching on the functionally graded plate resting on the elastic foundation is then considered,and the approach to shear functions is proposed to analyze the free vibration responses of the FGM plate resting on the Winkler/Pasternak/Kerr elastic foundation using the Reddy shear function and the new logarithmic shear function,respectively.The elastic foundation parameters are incorporated into the energy functional in the form of additional potential energy.The differential governing equations and boundary conditions of the FGM plate are derived by converting the dynamic problem into a problem of solving the eigenvalue equations based on the variational principle.The results show that the proposed logarithmic shear function is more accurate than the Reddy function.In addition,the effects of different functionally graded materials,power-law indexes,and geometric parameters on the natural frequency of the FGM plate resting on the elastic foundation are investigated.To address the problem that a large number of higher-order shear deformation theories are disordered and unrelated to each other,a unified model of displacement fields in the higher-order shear deformation theories is established according to the different types of transverse displacements and transverse shear strains.Based on the proposed unified model,,all existing displacement fields can be unified by setting the forms of transverse displacement and transverse shear strain,in addition,new displacement fields in the higher-order shear deformation theories can be established.The unified model of transverse shear functions is also introduced and a framework for modelling higher-order shear deformation theories is established.Based on the proposed unified model,bending,free vibration and buckling analysis of laminates and functionally graded plates are carried out,and the numerical results are compared with those in literature to verify the accuracy of the proposed new higher-order shear deformation theory.The results show that the unified model of higher-order shear deformation theory can theoretically cover all existing higher-order shear deformation theories,and the modelling framework can provide support for the modeling of functionally graded plates and shells.