Bending Analysis Of Functionally Graded Carbon Nanotube-Reinforced Composite Plates

Since the Carbon Nanotube(CNT)was first discovered by Japanese electron microscopist Iijima in 1991,interests in CNT has grown very rapidly in numerous scientific fields.CNT has unique and superior properties in mechanics,thermal and electricity aspects because of its large aspect ratio and specific surface area.Carbon Nanotube-Reinforced Composite(CNTRC)possessing more superior properties is naturally fabricated by introducing the CNT as ideal reinforcements into the advanced composite materials.However,due to the waviness and agglomeration effects of CNT as well as the weak bonding between CNT and matrix,the excessive carbon nanotube content may limit or even deteriorate the mechanical properties of the composite materials.To effectively improve the macroscopic properties of structures with a lower volume fraction of CNT,Functionally Graded Carbon Nanotube-Reinforced Composite(FG-CNTRC)which has a specific gradient distribution of CNT,was-first proposed by Shen at 2009.As the most basic structural elements in engineering,the beams,plates and shells made of this new composite material must have some unique advantages over traditional carbon nanotube-reinforced composite structures.Therefore,it is necessary and meaningful to carry out research on the mechanical behavior of FG-CNTRC beams,plates and shells.By now,the research on the mechanical behavior of FG-CNTRC beams,plates and shells mainly focused on the bending,buckling and vibration problems,and only theoretical and numerical studies were reported.To the author ’s best knowledge,the different problems’ governing equations of FG-CNTRC structures were derived based on either first-order or high-order shear deformation theory in almost all the literature,and all the nonlinear analysis of FG-CNTRC were based on the plate theories combined with von Karman assumption.In terms of plate and shell theory,besides the first-order or high-order shear deformation theory,some other experts assume that the transverse displacement is composed by two parts,bending and shear components,they are two independent physical entities.According to this viewpoint,a two-variable refined plate theory(RPT)was proposed by Shimpi.Compared to first-order or high-order shear deformation theory,RPT does not require any shear correction factor and has fewer unknown variables.Thai modified the RPT by adding an extra term in the transverse displacement to account for the thickness stretching effect.In terms of finite deformation theory,an exact unique decomposition theorem using tensor analysis and differential geometry method was proposed by Chen,named as Strain-Rotation(S-R)decomposition theorem.It is a revision of Stokes’ S-R decomposition theorem.The strain and local rotation occur simultaneously and without order,and the strain is uniquely determined.Due to its overcoming the deficiencies of classic finite defonnation theories,S-R decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation.In most cases,the analytical solutions are seldom obtained because of the complexity of nonlinearity problems.It is more likely for us to get the solutions in numerical ways.Being independent of the elements and meshes,element-free method has more advantages to solve large deformation problems compared to FEM.Therefore,a more reasonable and reliable geometric nonlinearity numerical method certainly will be established by combining the S-R decomposition theorem and element-free method.Based on the above knowledge,the new theory and method differing from existing work are employed to investigate the linear and nonlinear static bending behavior of FG-CNTRC rectangular plates.The main work is as follows:1.The modified,refined plate theory(RPT)accounting for thickness stretching effect is first employed to investigate the static small deformation bending behavior of FG-CNTRC plates.Governing equations are derived from the Hamilton principle.Closed-form solutions are obtained via the Navier solution.The effects of volume fraction of CNT and distribution forms of CNT on the bending behavior of CNTRC plates are discussed in detailed through numerical results.These two factors play a significant role in the bending deflection and stress of CNTRC plates.2.A three-dimensional geometric nonlinearity element-free Galerkin method(EFG)based on S-R decomposition theorem,named as 3D-SR-EFG is established.The incremental variation equation is derived from updated co-moving coordinate formulation and principle of potential energy rate,and three-dimensional discretization equations are obtained by element-free Galerkin method.By using the MATLAB programs based on the proposed 3D S-R element-free method in present study,the nonlinear bending problems for three-dimensional cantilever beam and rectangular plates(including simply supported isotropic,orthotropic and FGM plates and clamped orthotropic plates)subjected to uniform distributed load are numerical discussed.The reasonability,availability and accuracy of 3D S-R element-free method are verified through comparison studies.The effects on the convergence by size of sphere influence domain and penalty coefficients are studied in the nonlinear analysis of three-dimensional cantilever beam subjected to a uniform distributed load.3.The 3D-SR-EFG proposed in present work is employed to investigate the static large deformation nonlinear bending behavior of 4 types(UD,FG-V,FG-O,FG-X)of FG-CNTRC plates.Convergence and comparison studies for small deformation problems of FG-CNTRC are conducted to validate the numerical stability and accuracy of the proposed 3D-SR-EFG.The influences of volume fraction and distributions of CNT,plate’s aspect ratio and width-to-thickness ratio,boundary conditions on the nonlinear bending response of the CNTRC plates are numerically analyzed and discussed in parametric studies.Results demonstrate that volume fraction and distributions of CNT influence the nonlinear bending behaviour of CNTRC plates significantly.