| The composite material is composed of several different materials which can make full use of the advantages of performance from each other and produce synergistic effect.Compared with single phase materials,the composite material has more excellent mechanical properties and physical functions.Moreover,the composite material with different microstructures has different properties,such as high strength and light weight,wave propagation manipulation,shock absorption and noise reduction,etc.,which is widely used in the field of aerospace,vehicles,ships,functional devices,etc.In order to analyze and predict the performance of this kind of composite material,a method which can describe the microstructure characteristics of the composites is needed.Therefore,it is significant to study the high accuracy and efficiency numerical algorithm for this kind of composite materials.In addition,due to the evolution of biological materials,the mechanical properties are often optimized.Therefore,the study of biomimetic materials provide a reference and guidance for the design of materials.Based on the above background,this thesis has carried out four aspects of the static and dynamic problems and the wave propagation of three kinds of composite materials with microstructures,including the light layered materials,the bandgap materials and the nacreous bandgap materials.Firstly,according to the static and dynamic problems of the multilayered beam and plate structures,a multiscale static and dynamic algorithm is proposed based on the substructural boundary assumption.This algorithm uses the common base function to describe the fine-scale elements,and the higher order base function is used to describe the coarse-scale elements.The concept of the transformation matrix can be used to construct the multiscale boundary conditions conveniently and unifiedly.Because the degrees of freedom of the original problem are greatly reduced in the coarse-scale problem,the computational efficiency of the algorithm is greatly improved.In order to improve the accuracy of multiscale algorithm,the more complex or higher nonlinear interpolation function can be used to describe the coarse grid boundaries,including increasing the degrees of freedom in the nodes of coarse elements,increasing the degrees of freedom in the interior of coarse elements,increasing the boundary nodes in the coarse elements,such as the displacement function of the global-local higher-order theory.In addition,the efficiency,accuracy and convergence of the proposed algorithm with different base functions of coarse elements are also discussed.Moreover,in the dynamic response analysis,it is easy to implement the multiscale dynamic analysis by using the time integration algorithm in the equivalent static equilibrium equation.Finally,the accuracy and applicability of different boundary conditions are illustrated by the proposed multiscale numerical analysis of sandwich beams,multilayered beams and lattice structures.In the static and dynamic analysis,it also shows the rationality and efficiency of the proposed multiscale algorithm with the global-local higher-order theory.Secondly,according to the phononic and photonic bandgap composite materials,a new band structure and transmission characteristic analysis algorithm is proposed based on the multi-level substructure.For the band structure analysis problem,the Bloch boundary condition is applied after the finite element discretization,and the stiffness matrix of the internal degrees of freedom is independent of the reduced wave vector.Because of this feature,the internal degrees of freedom can be condensed to the Bloch boundary by the substructure and the traveling tree technique,and then the boundary conditions are applied before the generalized eigenvalue analysis.Therefore,in the process of each iteration,the global stiffness matrix factorization does not need to contain internal degrees of freedom,which can greatly reduce the size of the solution.Moreover,the influence of different initial vector selection methods on the computational efficiency is also discussed,and an efficient method for the selection of initial vectors is proposed.For the transmission characteristics,the bandgap materials have a finite size and some periodicity,each unit cell can be treated as the same substructure to carry out the translation transformation.The calculation efficiency can be improved,because the same substructure can only be calculated once.In addition,some classical examples are analyzed to discuss the accuracy,efficiency,convergence and memory usage of the algorithm,including three-dimensional locally resonant system,Lamb wave defect system,Bragg waveguide and the two-dimensional phoxonic crystal.The precision of the proposed algorithm is consistent with the traditional finite element method.Finally,a multi-objective topology optimization model is constructed by using the multi-level substructure algorithm and manufacturing constraints,which maximizes the relative phononic and photonic band gap and obtains a series of phoxonic crystal configurations with excellent performance.These results provide a good guidance and help for the design of the novel phoxonic crystal materials.Thirdly,according to the nacreous bandgap composite material with the special stacking microstructure,the tension-shear chain model and the finite element model of the elastic wave propagation are presented.The dispersion relation and energy attenuation of the one-dimensional tension-shear chain model are calculated by the transfer matrix method,and some interesting phenomena which are different from the traditional phononic crystals are found.A wide band gap range emerges in the low frequency region,and a very large relative band gap value is found.This means that the material can prevent the propagation of elastic waves in the wide frequency range.Then,two typical Brick and Mortar models are used todescribe the two-dimensional nacreous composite material,and the finite element models are established to analyze the band structure.The result of the multi-level substructure algorithm is consistent with the theoretical model.There are more abundant bands in the high frequency region of the band structure in the proposed algorithm which is different from the theoretical model,because the proposed algorithm can describe the higher order local shear modes in the mortar layer.Finally,the finite element models are established to analyze the transmission characteristics.It is found that there are wide ranges of elastic wave attenuation in the frequency range of the band gaps in both horizontal and vertical direction.The elastic wave attenuation is actually the reflection and the barrier to the elastic wave,which is different from the damping mechanism.In addition,nacre composite material does not appear the Fano-like interference phenomena which is found in the locally resonant phononic crystal,but with the similar lattice size.These features are beneficial to promote the application of low frequency broadband shock isolation and noise reduction.Finally,further research on the influence of the material and geometrical parameters on the band gap of the nacre with the special stacking microstructure is studied to better design and control the band gap of the material.In order to improve the efficiency,the multi-level substructure algorithm is employed.It is found that four kinds of parameters are sensitive to the band gap,including the mass density of brick,the elastic modulus of mortar,the length to height ratio and the connection length of brick and mortar.Then,the model of the super cell is established in order to study the influence of the defect on the band structure of the nacre,including the point defect,the line defect,the crack defect and the distribution of the random material.It is found that the band structure of the nacreous material is not sensitive to the defect under certain conditions.This means that the band gap of the nacreous material has a strong robustness and the width and range of the band gap are not easy to be affected by the defects,which is beneficial to the application of engineering vibration isolation and noise reduction.Finally,a three-dimensional vibration isolation material is designed by using the staggered stacking hard and soft materials.When the specific operating frequency range is needed in the application of vibration isolation,the cell size of the material can be enlarged or reduced in proportion.Through the numerical test in harmonic vibration environment and random vibration environment,it is find that the designed material shows very significant vibration isolation effect compared with other three kinds of reference materials,which is consistent with its own band gap characteristics. |
