复合材料周期结构数学均匀化方法的一种新型单胞边界条件

A novel unit cell boundary condition of mathematical homogenization method for periodical composite structure

数学均匀化方法是计算周期复合材料结构的有效方法之一,单胞边界条件施加的合理性直接决定了影响函数控制方程的计算效率和精度,进而影响均匀化弹性参数和摄动位移的计算精度。本文首先将单胞影响函数作为虚拟位移处理,给出了单胞在结构中真实的边界条件,结果表明,四边固支适合作为二维结构单胞边界条件;其次,针对二维结构提出了超单胞周期边界条件,有效提高了影响函数的计算精度,并使用与虚拟位移相对应的虚拟势能泛函验证超单胞周期边界条件的有效性;最后,利用数值分析验证多尺度渐进展开方法的计算精度,强调了二阶摄动的必要性。

Mathematical homogenization method(MHM)is one of the most effective methods for dealing with periodical structures,and its calculation accuracy and efficiency largely depend on the rationality of the unit cell boundary conditions which directly determines the accuracy of influence functions and perturbation displacements.In this paper,the influence function is treated as a virtual displacement firstly,and the real boundary conditions of a unit cell in the structure are obtained.The results show that clamped boundary condition is not fit as a boundary condition of a unit cell for a two-dimensional periodical structure.Secondly,for a two-dimension periodical structure,a super unit cell periodical boundary condition is proposed which effectively improves calculation accuracy of the influence functions and the virtual potential energy functional corresponding to the virtual displacement is proposed to verify the validity of the super unit cell’s periodical boundary condition.Finally,numerical analysis is utilized to verify the accuracy of the mathematical homogenization method in the case of a super unit cell boundary condition and the necessity of second-order perturbation is emphasized.