Bending,Buckling And Vibration Analysis Of Porous Beams

Porous materials have unique and excellent material properties.During the manufacturing process,one can disperse foams continuously and smoothly along one or more directions to adjust the local parameter of foams for achieving specific functions.Therefore,studying the mechanical properties of this functional material structure has important theoretical significance and application value.In this paper,the bending,buckling and vibration characteristics of porous beams(functionally graded microbeam,metal foam microbeam,metal foam core sandwich beam,three-dimensional graphene foam beam and three-dimensional graphene foam reinforced beam)are studied.The main work is as follows:(1)Based on the sinusoidal beam theory and the modified couple stress theory,bending and vibration analysis of functionally graded microbeams with porosity defect and subjected to thermo-hygro loadings is performed.The boundary conditions and governing equations of the microbeam are obtained by using Hamilton’s principle,and the analytical solutions of the deflection and natural frequency of the microbeam are given by Navier’s method.The results show that the existence of internal porosity defects and an increase in the porosity volume fraction result in the increase of the deflection of the microbeam.The influence of porosity distribution type and porosity volume fraction on the free vibration characteristics of the microbeam depends on the power-law index.As the power-law index,moisture change and temperature change increase,the deflection of the microbeam increases and the natural frequency decreases.(2)The bending and vibration of metal foam microbeams are studied.The microbeam model is established based on the sinusoidal beam theory and the modified strain gradient theory.The governing equations and boundary conditions of the microbeam are obtained using Hamilton’s principle,and the analytical solutions of the deflection and natural frequency of the microbeam are obtained using Navier’s method.The results show that the existence of foam increases the deflection of the microbeam and reduces the natural frequency.With the increase of foam coefficient,the deflection of the microbeam increases,and the natural frequency of the microbeam has different trends under different foam distribution types.Uniform distribution microbeams have the maximum deflection and the lowest frequency,and nonuniform distribution(dense at the top and the bottom,sparse at the mid-plane)microbeams have the minimum deflection and the highest natural frequency.(3)The free vibration of metal foam core sandwich beams embedded in Winkler-Pasternak elastic foundation is studied.The boundary conditions and governing equations of the structure are obtained using Timoshenko beam theory and Hamilton’s principle.The natural frequency of the sandwich beam is obtained using the Chebyshev collocation method.The results show that the Chebyshev collocation method has very high rate of convergence and achieves high precision.With the increase of core-to-face thickness ratio,the natural frequency of the sandwich beam increases initially and then decreases,and foam distribution plays more and more important effect on the vibration characteristics.The Winkler-Pasternak foundation makes the beam stiffer.Nonuniform distribution(dense at the top and the bottom,sparse at the mid-plane)beams have the maximum stiffness,and their natural frequencies increase with increasing foam coefficient;nonuniform distribution(sparse at the top and the bottom,dense at the mid-plane)beams have minimum stiffness,and their natural frequencies decrease as the foam coefficient increases;natural frequency of the uniform distribution beam is insensitive to changes in foam coefficient.(4)The bending,buckling and vibration of three-dimensional graphene foam beams are studied in the framework of sinusoidal beam theory.Graphene foam has three different distributions along the thickness of the beam.The governing equations and boundary conditions are derived based on Hamilton’s principle.The deflection,critical buckling load and natural frequency of the beam are calculated using Navier’s method and the Rayleigh-Ritz method.The results show that the nonuniform distribution(dense at the top and the bottom,sparse at the mid-plane)beams have the minimum deflection,the largest critical buckling load and the highest natural frequency;the nonuniform distribution(sparse at the top and the bottom,dense at the mid-plane)beams have the maximum deflection,the smallest critical buckling load and the lowest natural frequency.With the increase of the foam coefficient,the deflection of the beam increases,while its critical buckling load and natural frequency decrease,and the differences in beam deflection,critical buckling load and frequency between different foam distributions become more and more significant.(5)Using the Timoshenko beam theory and von Kármán type geometric nonlinear relationship to analyze the nonlinear free vibration of three-dimensional graphene foam reinforced beams.The governing equations of the beam are obtained by using the Ritz method,and the nonlinear frequency is obtained by the direct iterative method.The results show that as the vibration amplitude increases,the nonlinear/linear frequency ratio of the beam increases.Nonuniform distribution(dense at the top and the bottom,sparse at the mid-plane)beams have the highest nonlinear frequency and the minimum nonlinear/linear frequency ratio,and nonuniform distribution(sparse at the top and the bottom,dense at the mid-plane)beams have the lowest nonlinear frequency and the maximum nonlinear/linear frequency ratio.With the increase of foam coefficient,the nonlinear/linear frequency ratio of beams under different foam distributions has different trends.