| With the development of technology exploration, especially in the field of aerospace, the performance of high-tech equipment related to scientific research plays a key role, in which the material has a critical influence. Due to its extraordinary material behaviors, such as negative refractive index, negative equivalent mass density and negative equivalent modulus etc., metamaterials have attracted a lot of attention. In 1995, Milton proposed a new type of artificial meta-materials, and named it Pentamode Materials when he studied the material with extreme behaviors. Five eigenvalues of equivalent elastic modulus of this material are zero, and only one non-zero, so this material has only one load-carrying mode under static forces situation, and can only transmit one kind of elastic waves. With its excellent characteristics in terms of elastic wave propagation, Pentamode Materials obtaines a large number of applications in acoustic cloak. In order to control the propagation path of the elastic wave, the researchers hope to be able to design Pentamode Materials structure under the condition that nodes of structure connection topology and shape is constant.It should be noted that Pentamode Materials are novel artificial acoustic metamaterials, and they are usually designed by using solid materials, which microstructures are micro truss structures. However, the structures constructed by Pentamode Materials are theoretically unstable. The traditional criterion to judge the structure stability is based on the Maxwell criterion. Structural stability is determined by comparing the rob numbers of the structure and the structural degrees of freedom, when the rob number of the structure is greater or equal than the structural number of degrees of freedom, the structure is stable; otherwise, the structure is unstable. However, with the structure coupling topology and shape complexity increasing, relying solely on Maxwell criteria does not completely judge the stability of the structure. With the development of the equilibrium matrix and stiffness matrices theory, the researchers find that when the equilibrium matrix of micro-truss structure containes the movable displacement mode, if there is no external force, the structure is unstable; However, if there is external force, whether the structure that containing movable displacement model is stable, according to the elasticity theory, it should judge its stiffness matrix. If the stiffness matrix of structure is positive, the structure is stable; otherwise, the structure is unstable.From the perspective of global analysis of the entire structure, this paper proposes a method by employing preloads to insure the stability of the designed structures with Pentamode Materials, and a sufficient condition for the stability of structures with Pentamode Materials is presented. Meanwhile, two numerical examples of typical microstructures of Pentamode Materials are offered to verify the condition. Furthermore, in order to apply and implement the stability condition of structures in the structural design procedure, an optimization model of structures is established to describe the design problem, which views the structure design with Pentamode Materials as the transformation procedure of Pentamode Materials. As a result, if the unit cells of Pentamode Materials are constructed by micro trusses the optimization model is simple, both of the structure stability constraints and the characteristic constraints of Pentamode Materials are linear. Under the condition that nodes of structure connection topology and shape is constant, with the link length of micro-structures as optimization variables, the present study optimizes the shape and size of the typical 2-D and 3-D Pentamode Materials. This optimization model achieves the Pentamode Material with the best shapes under external loads. Under the equilibrium and stable condition of the structure, through iterating new node coordinates continuously, the best solution was reached in the feasible region. |
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