In this paper, a strain gradient continuum model for a metamaterial with a periodic lattice substructure is considered. A second gradient constitutive law is postulated at the macroscopic level. The effective classical and strain gradient stiffness tensors are obtained based on asymptotic homogenization techniques using the equivalence of energy at the macro- and microscales within a so-called representative volume element. Numerical studies by means of finite element analysis were performed to investigate the effects of changing volume ratio and characteristic length for a single unit cell of the metamaterial as well as changing properties of the underlying material. It is also shown that the size effects occurring in a cantilever beam made of a periodic metamaterial can be captured with appropriate accuracy by using the identified effective stiffness tensors.

本文考虑了具有周期性晶格子结构的超材料的应变梯度连续体模型。在宏观水平上假定了第二梯度本构定律。有效的经典和应变梯度刚度张量是基于渐近均匀化技术获得的，其中使用了所谓的代表性体积元内的宏观和微观能量的等效性。通过有限元分析进行了数值研究，以研究超材料的单个晶胞的体积比和特征长度的变化以及底层材料的性能变化的影响。