Beam model for the elastic properties of material with spherical voids

TL;DR

This paper introduces a beam network model to analyze the elastic properties of materials with spherical voids, especially at high void fractions, revealing anisotropic behavior and power-law relations for stiffness loss.

Contribution

A novel analytical beam model for predicting elastic properties of porous materials with high void fractions, including overlapping voids and anisotropic effects.

Findings

Power-law relation between Young's modulus and void fraction with exponents 5/2 and 3/2.

Good agreement between the beam model and finite element simulations.

Hexagonal close-packed voids exhibit highly anisotropic elastic behavior.

Abstract

The elastic properties of a material with spherical voids of equal volume are analysed using a new model, with particular attention paid to the hexagonal close-packed and the face-centred cubic arrangement of voids. Void fractions well above 74 \% are considered, yielding overlapping voids as in an open-cell foam and hence a connected pore structure. The material is represented by a network of beams. The elastic behaviour of each beam is derived analytically from the material structure. By computing the linear elastic properties of the beam network, the Young's moduli and Poisson ratios for different directions are evaluated. In the limit of rigidity loss a power law is obtained, describing the relation between Young's modulus and void fraction with an exponent of 5/2 for bending-dominated and 3/2 for stretching-dominated directions. The corresponding Poisson ratios vary between 0 and 1. With decreasing void fraction, these exponents become 2.3 and 1.3, respectively. The data obtained analytically and from the new beam model are compared to Finite Element simulations carried out in a companion study, and good agreement is found. The hexagonal close-packed void arrangement features very anisotropic behaviour, comparable to that of fibre-reinforced materials, This might allow for new applications of open-cell materials.