Coordinate Transformation Calculation Of Pentamode Metamaterial

Acoustic cloak, as a new branch of acoustics research, has become a research hotspot in recent years. Its basic theory is the coordinate transformation theory which is based on the form invariance of Maxwell’s equations under different coordinates. As a new acoustic metamaterials, the materials of acoustic cloak is synthetic and have unconventional material properties. The new type of acoustic metamaterials’pentamode material’(PM) has been researched in this thesis. The pentamode metamaterial is a new kind of synthetic material which is used in the acoustic cloak structure. The PM is named because its five of the six eigenvalues are zero. So the PM is rank one, for which the only stress-strain eigenmode is a hydrostatic stress or pure pressure and the five easy modes are all pure shear. Because of the particular characteristics of this PM, the study of the material structure is very valuable. This thesis has calculated the spatial coordinate transformation of PM. This provides a theoretical basis for the structural design of PM which is one of acoustic metamaterials in the future.Firstly, this thesis describes the development history of acoustic cloak and acoustic metamaterials, which leads to a spatial coordinate transformation for acoustics transform theory. Meanwhile, the development of PM and the material properties of PM have been introduced in this thesis. Some theoretical basis of this thesis is revealed in the first part, which include nonlinear finite elements for continua structures and Lagrangian method for solving large deformation, These provide a new method to calculate the coordinate transformation of PM structure.Secondly, we used the divergence equation of PM which was proposed by A.N.Norris and nonlinear finite element method to deduce the equilibrium equation for transformation of coordinates and the element stiffness matrix of the PM structure. The mathematical model of two-dimensional with four-node elements of PM structure has been worked out by the preparation of Matlab procedures. While this thesis compares with the analytical solutions and numerical solutions of standard circle domains of PM structure. These results prove the correctness of the equilibrium equation for transformation of coordinates and the element stiffness matrix of the PM structure.Finally, this thesis gives some calculation examples of transformation of coordinates of the PM structure, which based on the equilibrium equation and the element stiffness matrix.The first example is oval region whose inner boundary is circle. The second example is oval region whose inner boundary is oval. The third example is oval region whose inner boundary is square. The relevant structures have been analyzed, and get the deformation results of PM structure in the different parameters finally. These results provide a theoretical reference for coordinate transformation calculation of PM structure.