Study On Bending And Stability Of Functionally Graded Nanoplates And Graphene Foam Plates

In practical applications,plate components are usually subjected to in-plane compression loads and transverse loads,which may lead to bending and instability of plate structures.Therefore,the study of mechanical problems such as bending and stability of plate structures is particularly important in device design.Nanostructures have been widely used in many fields such as biology,medicine,micro-electro-mechanical systems and aerospace due to their excellent thermal,mechanical and electrical properties.Nowadays,engineers not only have higher and higher requirements for structural design,but also increasingly demand for new materials with high strength,light weight and high performance.For example,functionally graded materials,metal foam materials,three-dimensional graphene foam materials,threedimensional graphene foam reinforced materials and piezoelectric materials.Therefore,it is necessary to combine these materials with macro/micro structures,analyze and study their mechanical properties,and provide necessary technical support for device design in practical applications.This paper studies the bending and stability of functionally graded metal foam nanoplates,functionally graded piezoelectric nanoplates,three-dimensional graphene foam plates,and three-dimensional graphene foam reinforced plates.The main work is as follows:(1)The buckling of functionally graded metal foam nanoplates is studied based on nonlocal elastic theory and two-variable refined plate theory.The governing equations are derived by Eringen’s non-local constitutive relationship and Hamilton’s principle.Analytical solutions for the buckling load of functionally graded metal foam nanoplates are obtained by the Navier method.The results show that the functionally graded metal foam nanoplate has a smaller critical buckling load than its solid metal nanoplate;as the foam coefficient,small-scale parameters and aspect ratio increase,the critical buckling load of the nanoplate decreases;Under the larger mode number n,the effect of small-scale parameters on the buckling load is very significant.(2)The non-local elastic theory and two-variable refined plate theory are used to study the dynamic stability of functionally graded piezoelectric nanoplates.Based on Eringen’s nonlocal constitutive relationship of piezoelectric materials and Hamilton’s principle,the governing equation is deduced.The Mathieu-Hill equation is obtained using the Navier method.Finally,the unstable region of the system is determined by using the Bolotin method.The results show that as the small-scale parameters increase,the origin of the unstable region of the functionally graded piezoelectric nanoplate is shifted to a lower excitation frequency;compared with the power law index and small scale parameters,the influence of the change of external voltage on the unstable region of nanometer plate is more significant.;under a given dynamic load factor,the unstable region of the nanoplate with larger length-to-thickness ratio is narrow,and it is prone to dynamic instability at lower excitation frequency.(3)The bending and buckling of three-dimensional graphene foam plate are studied.The material properties of three-dimensional graphene foams vary continuously along thickness direction of the plate.Consider three types of foam distribution along the thickness direction.The governing equations are obtained by using the Hamilton’s principle,and the analytical solutions of the plate deflection and stress are obtained by using the Navier method.In addition,the buckling of 3D graphene foam plates under various boundary conditions is studied by using the Galerkin method.The results show that the larger the foam coefficient,the larger the central deflection of the three-dimensional graphene foam plate,and the smaller the critical buckling load;plates with larger foam coefficients or smaller aspect ratio,the effect of foam distribution on central deflection and critical buckling load is more obvious;at smaller axial compression ratios,the foam distribution type has a significant effect on the critical buckling load.(4)In the framework of Kirchhoff plate theory,combined with von Kármán geometric nonlinear relationship,the nonlinear dynamic stability of three-dimensional graphene foam reinforced plates is studied.The Galerkin method and Airy stress function are used to obtain the nonlinear Mathieu-Hill equation of the system.The Bolotin method is used to obtain the stable and unstable solutions of the transverse vibration amplitudes of the system.The results show that the transverse vibration amplitude of the three-dimensional graphene foam reinforced plate gradually decreases with the increase of the length-to-thickness ratio La/h or the lengthto-width ratio La/Lb;the plate instability excitation area increases as the static load factor increases;the larger the dynamic load factor β,the larger the instability excitation area of the plate,the smaller external load safety excitation frequency,and the transverse vibration amplitude of the system increases with the increase of the dynamic load factor β.