Effective in-plane stiffness of unidirectional periodic nanoporous materials with surface elasticity

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In this paper, the in-plane effective elastic properties of a porous material with periodic nanoscale holes of arbitrary shapes are investigated. A square representative unit cell (RUC) with a central hole is analysed for the original structure. On the edges of the RUC, proper periodic displacement boundary conditions are imposed. On the surfaces of the holes, the stress boundary condition is formulated through the Gurtin–Murdoch surface elasticity model. The problem is finally solved via the complex variable techniques, like superposition principle, conformal mapping, series expansion and collocation methods. Numerical examples for two common shapes, circle and square, of holes are presented. The results show that the in-plane effective properties of the structure are significantly influenced by the volume fraction (VF), size, shape of the holes, along with the surface elasticity. Specifically, a decrease in the VF or an increase in the size of the holes can lead to an increase in most of the effective moduli of the structure; a structure with circular holes has overall larger effective moduli than a structure with square holes; the surface elasticity can play a dominant role for certain effective moduli.