| Cellular solids have the advantages of light weight,high strength,and high specific stiffness.It is of great significance to study the design method of cellular solids.According to the geometric dimensions,the design of cellular solids can be divided into two levels those are structural design and unit cell design.The former studies the structural configuration consisting of fixed cellular solids;and the latter studies the unitary configurations required to fill a fixed structure.Structural topology optimization method is an effective way to solve these two types of design problems.In the topology-optimized design of cellular solid cells,it is generally assumed that the size of the unit cells is much smaller than the size of the macrostructure,and the cellular solid is equivalent to a classical continuous medium with an equivalent material constant.This assumption ignores the effect of the size of the structure or component on macro performance.However,the minimum size of the cellular solid structure is often similar to the size of the unit cell,so that the dynamic and static mechanical response of the macrostructure is dependent on the size of the unit cell.Thus,the design of the unit cell cellular solid of such size effect needs to be considered.In order to describe the dependence of the mechanical behavior of cellular solid structures on the unit cell size,this paper homogenized the cellular solids as couple stress continuum with equivalent material constants,based on the couple stress theory.By optimizing the design of the unit cells of the periodic cellular solid,the free vibration characteristics of the macrostructure are improved.Specifically,the following work are carried out:1.Aiming at the existence of a large number of cellular solid beam structures in actual engineering,a topological optimization method for the design of such special cellular solid unit cells was developed.An optimization model is established.The objective function is to maximize the free vibration fundamental frequency of the structure,the constraint function is the volume fraction of the periodic cellular solid,and the design variable is the cell density variable of the unit cell;Through the analysis of unit cells,the material constants of the continuum of equivalent couple stress of cellular solids were solved.Using the theory of couple stress media beam,the natural frequency of the cellular solid beam structure was analytically calculated,which greatly improved the calculation efficiency.2.The finite element formulation of the free vibration problem of the couple stress medium was studied.In the framework of the couple stress theory,Hamiltonian principle is used to derive the free vibration equation of the coupled medium,and the finite element discrete form is given.Relying on the idea of discrete-point constraints in finite element analysis of thin plates,set up a continuum of couple-stress units that satisfy C1 with discrete-point constraints,and used to solve free vibration problems.With typical examples,the accuracy of finite element formulas was verified by the comparative analysis of finite element results and analytical results.3.A topological optimization model for the unit cell design with the maximum fundamental frequency of any periodic cellular solid structure was established.The design variable of the optimization model is the density variable of the unit cell,and the volume constraint is the volume fraction of the cellular solid.During the optimization process,the equivalent constitutive parameters and inertial parameters of the equivalent couple stress medium of the cellular solid were established through the analysis of the unit cell,and the relationship between the cellular solid structure and the unit cell was related.The free vibration frequency of the structure was calculated by using the finite element method,the sensitivity expression of the objective function was deduced by the analytical method,and the optimization problem was solved by the feasible direction method.In the optimization design process,the sensitivity filtering method was used to eliminate the grid dependence of the results and the checkerboard format. |
