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Research Papers

# Impact of the Lattice Angle on the Effective Properties of the Octet-Truss Lattice Structure

[+] Author and Article Information
Mohamed Abdelhamid

Department of Mechanical Engineering,
Lassonde School of Engineering,
York University,
4700 Keele Street,
e-mail: mahamid@yorku.ca

Aleksander Czekanski

Department of Mechanical Engineering,
Lassonde School of Engineering,
York University,
4700 Keele Street,
e-mail: alex.czekanski@lassonde.yorku.ca

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received October 1, 2017; final manuscript received May 23, 2018; published online June 22, 2018. Assoc. Editor: Huiling Duan.

J. Eng. Mater. Technol 140(4), 041010 (Jun 22, 2018) (11 pages) Paper No: MATS-17-1287; doi: 10.1115/1.4040409 History: Received October 01, 2017; Revised May 23, 2018

## Abstract

Cellular materials are found extensively in nature, such as wood, honeycomb, butterfly wings, and foam-like structures like trabecular bone and sponge. This class of materials proves to be structurally efficient by combining low weight with superior mechanical properties. Recent studies have shown that there are coupling relations between the mechanical properties of cellular materials and their relative density. Due to its favorable stretching‐dominated behavior, continuum models of the octet‐truss were developed to describe its effective mechanical properties. However, previous studies were only performed for the cubic symmetry case, where the lattice angle $θ=45$ deg. In this work, we study the impact of the lattice angle on the effective properties of the octet-truss: namely, the relative density, effective stiffness, and effective strength. The relative density formula is extended to account for different lattice angles up to a higher-order of approximation. Tensor transformations are utilized to obtain relations of the effective elastic and shear moduli, and Poisson's ratio at different lattice angles. Analytical formulas are developed to obtain the loading direction and value of the maximum and minimum specific elastic moduli at different lattice angles. In addition, tridimensional polar representations of the macroscopic strength of the octet‐truss are analyzed for different lattice angles. Finally, collapse surfaces for plastic yielding and elastic buckling are investigated for different loading combinations at different lattice angles. It has been found that lattice angles lower than $45$ deg result in higher maximum values of specific effective elastic moduli, shear moduli, and strength.

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## Figures

Fig. 2

A comparison between relative density values using Eq.(1) versus computer-aided design (CAD)-extracted values at different lattice angles

Fig. 1

The octet-truss unit cell with the tetrahedron substructure geometry (shown in darker color) and the transformation coordinate systems (numbers shown identify the six tetrahedron truss members)

Fig. 3

Tridimensional polar representations of the effective specific elastic modulus for different lattice angles: (a) orientation of the octet-truss, (b), (c), and (d) represent the specific stiffness for θ=40 deg,45 deg, and 50  deg, respectively

Fig. 7

Tridimensional polar representations of the effective specific strength for different lattice angles: (a) orientation of the octet-truss, ((b), (c), and (d)) represent the specific strength for θ=40 deg,45 deg,and  50  deg, respectively

Fig. 4

Behavior of  cos α2 against the lattice angle θ

Fig. 5

Specific stiffness and percentage relative density of the octet-truss lattice with respect to the lattice angle θ for two aspect ratios r/L=0.07 and 0.14

Fig. 8

Collapse surface of the octet-truss due to plastic yielding in (σ¯xz,σ¯zz) space (a) and (σ¯xx,σ¯yy) space (b) in specific strength formulation at an aspect ratio of r/L=0.14

Fig. 6

Three-dimensional Cartesian representations of the transformed specific shear moduli for θ=45 deg

Fig. 9

Collapse surface of the octet-truss due to elastic buckling in (σ¯xz,σ¯zz) space at an aspect ratio of r/L=0.14 and a yield strain of εy=0.1 (a) and εy=0.05 (b)

Fig. 10

Collapse surface of the octet-truss due to elastic buckling in (σ¯xx,σ¯yy) space at an aspect ratio of r/L=0.14 and a yield strain of εy=0.1 (a) and εy=0.05 (b)

## Errata

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