Motivated by experimental findings on deformation induced microcracks in thin metal films and by their influence on the effective macroscopic electrical conductivity, a computational multiscale formulation for electrical conductors is proposed in this contribution. In particular, averaging theorems for kinematic quantities and for their energetic duals are discussed, an extended version of the Hill–Mandel energy equivalence condition is proposed and suitable boundary conditions for the microscale problem are elaborated. The implementation of the proposed framework in a two-scale finite element environment is shown and representative boundary value problems are studied in two- and three-dimensional settings.

受实验结果的启发，在金属薄膜中由变形引起的微裂纹及其对有效宏观电导率的影响，提出了一种用于计算电导体的多尺度计算公式。特别是，讨论了运动量及其能量对偶的平均定理，提出了Hill-Mandel能量当量条件的扩展版本，并阐述了微尺度问题的合适边界条件。示出了在二维尺度有限元环境中所提出的框架的实现，并且在二维和三维环境中研究了代表性的边值问题。