Finite-volume homogenization and localization of nanoporous materials with cylindrical voids. Part 1: Theory and validation

Abstract

Abstract Surface elasticity effects based on the Gurtin-Murdoch model are incorporated for the first time into a finite-volume based homogenization theory to enable analysis of materials with nanoscale cylindrical voids of circular and ellipsoidal cross-section in periodic arrays. In a departure from the previously employed enforcement of traction and displacement continuity conditions in a surface-average sense applied locally to each subvolume of the unit cell, the Young-Laplace equilibrium equations are implemented using a central-difference approach involving adjacent subvolumes, an approach both new to the finite-volume theory as well as necessary. In Part 1, the new computational capability is validated by published results on homogenized moduli, stress concentrations and full-field stress distributions in nanoporous aluminum obtained using elasticity-based and numerical approaches. Notably, numerical problems associated with singular-like stresses and associated instabilities experienced in finite-element solutions (as well as the elasticity solution of an elliptical void in an infinite matrix) are not as pronounced in the proposed approach, enabling determination of surface and full-field stresses in a wider range of pore radii. New results are generated in Part 2 aimed at demonstrating the effects of nanopore array type and aspect ratio of elliptical voids on homogenized moduli and local stress fields in a wide range of porosity volume fractions and radii. These results highlight the importance of adjacent pore interactions neglected in the classical micromechanics models , that remains to be quantified by numerical homogenization techniques.