Shear Stiffness Prediction Of Periodic Beam And Plate Structures And Two-scale Concurrent Topology Optimization

Beam and plate structures with periodic microstructures are widely used in industrial applications due to their excellent performances.For numerical analysis of their mechanical performance,these structures are usually homogenized as homogeneous beams and plates with effective stiffness,due to the excessively high computational cost and time in direct numerical analysis.Asymptotic homogenization(AH)theory is an effective method in predicting effective properties of these periodic heterogeneous structures with rigorous mathematical foundation,while the lack of direct numerical implementation hinders its extensive application.To overcome this difficulty,Gengdong Cheng et al.propose a novel numerical implementation of asymptotic homogenization method(NIAH).A drawback of AH method is that it cannot take transverse shear deformation into account and can only homogenize periodic beams and plates into Euler-Bernoulli beams and Kirchhoff plates,which may cause significant error for short beams and thick plates.Therefore,shear stiffness prediction of periodic beams and plates,along with homogenization techniques of these structures into homogeneous Timoshenko beams and Mindlin plates to improve the prediction accuracy of their mechanical responses,receives sustained academic attention.This dissertation presents a systematic method for prediction of shear stiffness and relevant mechanical responses of periodic plates and beams based on the intuitionistic interpretation of NIAH method.Topology optimization of structures with periodic microstructures can improve structural performance and achieve new structural configurations,which is a hot topic in topology optimization.The two-scale concurrent topology optimization approach optimizes both macro material layout and micro base cell configurations to promote structural performance,which is very important in practical applications.Two scale concurrent topology optimization of periodic plates and periodic structures with multiple materials is studied in this dissertation.The main content of this dissertation is as follows:(1).This dissertation further extends NIAH approach.A simplified solution procedure of NIAH method is proposed to improve solution efficiency and an intuitionistic interpretation of NIAH method is introduced.The parallel axis formulae of effective stiffness of periodic beams and plates are derived which establishes the relation between the effective stiffnesses in original and new coordinates.It is also illuminated in this dissertation that the application of unit shear strain cannot be utilized to evaluate effective shear properties of periodic beams and plates in AH framework.New approaches should be developed for effective shear stiffness prediction.(2).This dissertation proposes a systematic FE-based approach for effective shear stiffness prediction of periodic beams.By matching the internal and external force state of a macro homogeneous beam and that of a micro unit cell,a stress and strain state of the unit cell,which corresponds to the pure shear state of the macro beam,is established and the formulation of effective shear stiffness is derived through energy equivalence.The FE formulation of effective shear stiffness is constructed and its efficient numerical implementation is achieved.By using the effective shear stiffness and the Timoshenko beam theory,the displacement and stress,especially the shear stress of the periodic beam is predicted with high accuracy.(3).This dissertation proposes a systematic FEM-method based approach of effective shear stiffness prediction of periodic plates.By matching the internal and external force state of a macro homogeneous plate and that of a micro unit cell,a stress and strain state of the unit cell,which corresponds to the pure shear state of the macro plate,is established and the formulation of effective shear stiffness is derived through energy equivalence.The FE formulation of effective shear stiffness is constructed and its efficient numerical implementation is achieved.The displacement response of the original periodic beam is predicted with the new homogenization approach,and numerical results show that,compared to Kirchhoff plate model,Mindlin plate model can substantially increase displacement prediction accuracy.(4).This dissertation studies two-scale concurrent topology optimization of periodic plates for maximum eigenvalue buckling load.The analytical sensitivity of eigenvalue buckling load with respect to macro and micro design variables is derived.With the aid of NIAH approach,the objective function and sensitivity are partly calculated from output data of FE analysis,which simplifies the solution procedure and improves efficiency.Numerical results show that,compared to solid plates,optimized porous or stiffened plates can significantly raise the eigenvalue buckling load,which illustrates the advantage of two scale concurrent topology optimization.(5).This dissertation studies two scale concurrent topology optimization of structures with multiple heterogeneous materials.Two-scale optimization with different materials in different prescribed domains is first studied.For structures with complicated stress states in macro scale,two scale topology optimization based on macro principal stress orientation is further researched.By penalizing the volume of materials that don't comply with usage criterion and constraining the total volume fraction of penalized materials,a simple optimization formulation that is efficient in material classification is established.Numerical results show that compared to the former material classification approach,the latter one can achieve better optimal results and structural performance.