In this paper, size effects exhibited by mechanical metamaterials have been studied. When the sizescale of the metamaterials is reduced, stiffening or softening responses are observed in experiments. In order to capture both the stiffening and softening size effects fully, a second-order asymptotic homogenization method based on strain gradient theory is used. By this method, the metamaterials are homogenized and become effective strain gradient continua. The effective metamaterial parameters including the classical and strain gradient stiffness tensors are calculated. Comparisons between a detailed finite element analysis and the effective strain gradient continua model have been made for metamaterials under different boundary conditions, different aspect ratios, different unit cells (closed or open cells) and different topologies. It shows that both stiffening and softening size effects can be captured by using the effective strain gradient continua models.

在本文中，已经研究了机械超材料所表现出的尺寸效应。当超材料的尺寸减小时，在实验中观察到硬化或软化响应。为了充分捕捉到刚度和软化尺寸的影响，采用了基于应变梯度理论的二阶渐近均匀化方法。通过这种方法，超材料被均质化并成为有效应变梯度连续体。计算有效的超材料参数，包括经典和应变梯度刚度张量。对于超材料在不同的边界条件，不同的纵横比，不同的单位单元（闭合或开放单元）和不同的拓扑结构下，已经进行了详细的有限元分析和有效应变梯度连续模型的比较。