| Pentamode metamaterials (PM materials) is a new kind of artificial materials, and their main characteristics are that only one eigenvalue of elastic modulus matrix of the Pentamode metamaterials is not zero, so that Pentamode metamaterials have only one stress mode and they can be described by the bulk modulus, which makes Pentamode metamaterials behave like a fluid. As a results, Pentamode metamaterials show big potential in the design of acoustic cloaking. However, according to A. N. Norris’s theory, the design for Pentamode metamaterials is different from other metamaterials, because a nonlinear partial differential equation must be satisfied while using the coordinate transformation method to design Pentamode metamaterials. The thesis derives the solutions of the partial differential equation using total Lagrangian method and provides a reference for the design of Pentamode metamaterials.Firstly, a brief introduction to the acoustic cloaking is given, then the development of acoustic transformation and metamaterials are introduced, which are the theoretical basis and material basis of acoustic cloaking respectively. Thus, the characteristics, the theory development and experimental progress of Pentamode metamaterials are elaborated in detail. To solve the nonlinear equation that Pentamode metamaterials satisfy, the concepts of continuum mechanics and the total Lagrangian methods for finite element analysis are also elaborated, which lays the foundation for the proposed approach.Secondly, the weak form of the nonlinear equation is proposed using total Lagrangian methods, then the weak form is descreted by finite element methods, thus, the nonlinear finite element formulation of the equation are elaborated and the iterative solution algorithm is derived.Finally, according to the proposed approach,2D and 3D numerical examples are given using MATLAB software, and the results show that the proposed approach is available for designing Pentamode metamaterials and acoustic cloaking. |
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