This paper aims to determine the in plane effective elastic properties of two-dimensional (2D) materials containing nanovoids of arbitrary shape. To achieve this objective, the complex variable and the associated conformal mapping techniques are used to solve the heterogeneity problem of a single nanovoid with arbitrary shape embedded in an infinite matrix. In this particular problem, to capture the edge and size effects of nanovoids, line elasticity model is used for the void boundary. The results of the heterogeneity problem are then used to determine the elastic properties of 2D nanoporous materials by applying the dilute and Mori–Tanaka schemes. Applications to the case of aluminum and the study the shape and size effects are also presented.

本文旨在确定包含任意形状纳米空隙的二维（2D）材料的面内有效弹性特性。为了达到这个目的，复杂变量和相关的共形映射技术被用来解决在无限矩阵中嵌入任意形状的单个纳米空隙的异质性问题。在这个特定问题中，为了捕获纳米空隙的边缘和尺寸效应，将线弹性模型用于空隙边界。然后，通过应用稀薄方案和Mori–Tanaka方案，将异质性问题的结果用于确定2D纳米多孔材料的弹性特性。还介绍了在铝外壳上的应用以及对形状和尺寸影响的研究。