基于渐近分析的均匀化方法确定超材料参数
Determination of Metamaterial Parameters by Means of a Homogenization Approach Based on Asymptotic Analysis
摘要：
由于增材制造技术，可以通过使用微米长度尺度（微米尺度）的晶格子结构来生产毫米长度尺度（宏观尺度）的结构。这种系统在宏观尺度上被称为超材料，因为其机械特性偏离微观尺度上的特性。作为一种补救措施，超材料是通过使用额外的参数建模;我们打算确定它们。基于渐近分析的均匀化方法在微观和宏观尺度上建立了这些不同特征之间的联系。在微观尺度上的线弹性一阶理论与在宏观尺度上的线弹性二阶理论相关。在宏观尺度上的参数关系是通过使用在一个所谓的代表性体积元（RVE）内的宏观和微观尺度上的能量的等效性而导出的。参数的确定是通过用有限元法（FEM）求解边值问题来完成的。所提出的方法保证了额外的参数消失，如果材料是纯均匀的，换句话说，它是完全兼容的传统的均匀化方案的基础上空间平均技术。此外，所提出的方法是可靠的，因为它确保了这种解析的附加参数对RVE的选择不敏感，而取决于结构的固有尺寸，RVE的选择包括较小的RVE的重复。
英文摘要：
Owing to additive manufacturing techniques, a structure at millimeter length scale (macroscale) can be produced by using a lattice substructure at micrometer length scale (microscale). Such a system is called a metamaterial at the macroscale as the mechanical characteristics deviate from the characteristics at the microscale. As a remedy, metamaterial is modeled by using additional parameters; we intend to determine them. A homogenization approach based on the asymptotic analysis establishes a connection between these different characteristics at micro- and macroscales. A linear elastic first order theory at the microscale is related to a linear elastic second order theory at the macroscale. Relation for parameters at the macroscale is derived by using the equivalence of energy at macro- and microscales within a so-called Representative Volume Element (RVE). Determination of parameters are succeeded by solving a boundary value problem with the Finite Element Method (FEM). The proposed approach guarantees that the additional parameters vanish if the material is purely homogeneous, in other words, it is fully compatible with conventional homogenization schemes based on spatial averaging techniques. Moreover, the proposed approach is reliable as it ensures that such resolved additional parameters are not sensitive to choices of RVE consisting in the repetition of smaller RVEs but depend upon the intrinsic size of the structure.
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