Design Of Material With Phononic Band-gap/Prescribed Properties

The materials are often required to exibit presicribed/specific and excellent properties inadvanced engineering. In some cases, these requirements may not be fulfilled by existedmaterials found in nature and developed in engineering. To design and fabricate newmaterials has been an important subject. The traditional and important method for newmaterial development is to arrange the distribution of atoms by chemical means. Thedesignable character of composite materials provides a new alternative method to developnew materials. The properties of composite materials consisting of multi-phases aredetemined by the representative parameters of microstructures (the base materials' properties,distribution, volume fraction etc.), and can be adjusted by changing the representativeparameters of microstructure. Microstructure topology design becomes an important methodto design new materials.In this dissertation, the propagation of elastic wave in truss-like material and band-gapcharacteristic are investigated, and elastic wave band-gaps are found in truss-like materialswith specific microstructures. Furthermore, the design optimization problem of truss-likematerials with widest phononic band-gaps is formulated and the solving algotithms aredeveloped. The designs of materials with zero thermal expansion and prescribed mechanicalproperties (such as minus Possion's ratio) are studied, a topology description function (TDF)method for designing the microstructures of materials with prescribed propertieds is presentedin this dessertation. The main content and results are given in the following paragraphs.1) Analysis of propagation of elastic wave in truss-like material and band-gapcharacteristic Truss-like material has high stiffness-weight and strength-weight ratio, so itcan be used as an utral-light material which has great potential applications in aeronautic andastronautic industry, automobile industry and others. It benefits the design of multi-functionalmaterial to analyze the propagation of elastic wave in periodic truss-like material and find thephononic (elastic wave or acousitic wave) band-gaps. In this paper the analytical solution oflongitudinal oscillation of bar is used as shape function relating the displacements of anypoint in a bar with that of the beginning and the end nodes. The wave equation of elastic waveis obtained based on the displacement of nodes. Based on plane expansion method andFloquet-Bloch theory, dispersion equation is obtained based on displacement of nodes in basecell. The solving algorithm of the dispersion equation which determines the band-gap is developed. The results show that there are phononic band-gaps in truss-like material and theband-gaps are verified by numerically simulated experiments which are implemented byusing CAE software. The topology, shape and size of microstructure have great effect on thewidth and position of band-gaps. To increase the efficiency, the propagation characteristic ofelastic wave in truss-like material is studied based on mass-spring model. The calculationprecision of mass-spring model is analyzed. It shows that in the range of low frequency thedispersion diagram obtained based on mass-spring model has acceptable precision and thecorresponding Calculation has high efficiency.2) Design optimization of truss-like material with band-gaps The photonic(electromagnetic wave) band-gap has a great deal of study, which appears at the range of highfrequency. The phononic (elastic wave or acoustic wave) band-gap materials have greatpotential applications such as sound or vibration protection devices. Hence, development of asystematic method for the design of truss-like material with the widest band-gap or band-gapat prescribed (or as low as possible) frequency range has great values in application andacademic study. In this dissertation, the design of phononic band-gap is studied. By choosingthe cross-sectional areas as design variables and the maximum width of band-gap as objective,the formulation and solving algorithm of design optimization are achieved. The effection ofshear stiffness on phononic band-gap is studied. The truss-like material with prescribed shearstiffness and the widest band-gap between adjacent bands is designed.3) Design of material with zero thermal expansion coefficients It is necessary todesign the material with zero thermal expansion coefficients since thermal stability is strictlyneeded in some instruments such as satellite, antenna, precision machining. In thisdissertation the design optimization of microstructure with zero thermal expansioncharacterisitic is formulated. Based on the idea of topology optimization, the volume fractionof every phase material is chosen as design variable, the property of zero thermal expansion isthe objective. The new microstructure of materials with zero thermal expansion coefficients isobtained, which consists of the three-phase isotropic material. The dependence on initialdesign is investigated, and some possible reasons causing the problem are discussed. Thethermal expansion behavior of the material consisting of designed microstructures is tested bynumerically simulated experiments and the thermal expansion coefficients are obtainedthrough analyzing the deformation of the materials caused by temperature change. Thesenumerically simulated experiments verify the characteristic of zero (low) thermal expansionbehavior of the designed material.4) Topology description function (TDF) based method for design optimization ofmaterials with prescribed properties The parameter description of topology is a key fortopology optimization. The topololgy can be effectively described by the volume fraction (or relatice density) of material in elements which are used to calculate the response of structureby finite element method (FEM), and this description method has been successfully applied intopology optimizaiton; however the design variables and results of optimization depend onelements of FEM. In this paper the formulation of designing microstructure with prescribedproperties is studied based on topology description method (TDF). By presenting the TDF asthe sum of a series of base functions, the topology of microstructure can be adjusted by theparameters of base functions. In this method, the topology optimization of microstructure isformulated as a size (or parameter) optimization problem whose design variables areparameters of base functions of TDF and independent on the mesh of the design domain. To acertain extend, the problem of dependence of optimization results on mesh is overcomed. Thismethod can determine a high quality topology for describing the distribution of constituentmaterials in design domain and avoid the checkerboard problems often met in topologyoptimization method based on mircostructure, such as SIMP method. Compared withconventional level set method, TDF method can be solved by existing optimizationtechniques and avoid solving the "Hamilton-Jacobi"-type equation by difference method. Thedesigns of base cells, which have big positive or negative Poisson's ratio, show that themethod based on TDF is effective for material design.The work of this dissertation is supported by the National Natural Science Foundation ofChina through the Grant No.s (10572030, 10332010, 10421202), the National Basic ResearchProgram of China (No.2006CB601205) and the program for new century excellent talents inuniversity of china (2004).