Analytical models of the geometric properties of solid and hollow architected lattice cellular materials
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New closed-form analytical equations for volume fractions and surface-area-to-volume ratios for architected lattice cellular materials are derived. Prior approximate equations which erroneously over count overlapping volumes and the associated surface area are commonly used in the literature. These equations are found to have up to 184% error for volume fraction calculations for hollow lattices and 211% error for surface-area-to-volume ratio calculations, thus necessitating computational methods to arrive at accurate geometric properties for cellular lattice materials. This work derives new equations which are accurate to better than 1% for both volume fraction and surface-area-to-volume ratio as compared to the computational models. These new equations for cellular lattice materials are applicable to both pyramidal and tetrahedral unit cells as well as to both hollow and solid lattice members. By eliminating the need for numerical models to compute accurate volume fractions and surface-area-to-volume ratios of architected cellular materials, these new analytical equations will enable accurate yet computationally efficient optimization of the physical properties of architected cellular materials.
I. INTRODUCTION
Cellular materials are porous materials with lower bulk density than that of their constitutive materials. These materials tend to be heavily interconnected with a high surface area (SA) and low density. They are useful in a wide variety of applications,1 such as sandwich structure cores,2,3 energy absorption,4,5 thermal management,6–8 bio scaffolds,9 and acoustics.10 The performance of cellular materials is dominated by three major factors: material, architecture, and relative density.11 To properly design a cellular material for a given application, each of these three factors must be determined. The base material choice is important since the properties of the constituent solid are extended to the individual members that make up the porous material. The architecture dictates the behavior of the porous material as a whole and includes open or closed cell pore types and stochastic or periodic cell structures.12 Finally, the relative density is of paramount importance because many properties scale according to a power-law relationship with relative density, depending on the architecture.13 Many types of architectures exist for cellular materials. Stochastic foams have been in production for decades and the physical properties of many of these porous Contributing Editor: Lorenzo Valdevit a) Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2017.427
materials, both open cell and closed cell, have been extensively investigated and characterized.14–16 While stochastic foams excel in energy absorption due to their near-constant plateau stress,15 the random nature of the foam results in a bending-dominated structure that has decreased stiffness.17 Furthermore, while the bulk material properties of these foams can be tailored for a specific application, the random nature of the s
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