^{[Photo by Basile Ugnon-Coussioz via Unsplash]}

Honeycombs have fascinated humans for over two centuries. Greek mathematician Pappus of Alexandria (290–350 CE) proposed that a hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter, which suggests a way to build the widest structure with the least amount of material. Honeycomb structures have been used in many applications, including as a constructional element in 3D printing. In general, grids (rectangular, square, hexagonal or triangular) are found naturally, and we find materials which are nothing but rows and rows of such grids, called ‘lattices’, made up of atoms or molecules. An example is graphene, a form of carbon. In this article, we will focus on such grids that are hexagonal or honeycomb lattices.

A group of researchers from the Indian Institute of Technology Bombay (IIT Bombay), the Indian Institute of Technology Kanpur (IIT Kanpur), and Swansea University, United Kingdom, have studied how lattices affect the properties of materials that we see around us. Their results could be useful in expanding the scope of 3D-printing materials with desired mechanical properties in different directions. The findings have been published in the journal *Extreme Mechanics Letters*. The research was partially supported by IIT Kanpur, IIT Bombay, and the European Commission.

When a material’s properties, like elasticity, do not depend upon the direction in which it is studied, physicists call the property ‘isotropic’. On the other hand, when the property depends on the chosen direction, it is said to be ‘anisotropic’, whose degree may vary depending on the lattice structure. In the current study, the researchers theoretically analysed how the properties of naturally-occurring as well as engineered materials depend on the intrinsic geometry of the lattice.

“We systematically explored the theory linking the relationship between isotropy or anisotropy and the lattice geometry,” says Prof Susmita Naskar of IIT Bombay, a co-author of the study.

The researchers focussed on properties of solids, like the elasticity, which quantifies how much a material resists deformation when subjected to external forces. By solving the equations that determine the behaviour of the lattices, they investigated how the overall elasticity of the material depends on the geometry of the lattice. Then, they characterised the mathematical dependence of the elasticity of the material on the lattice structure and its geometrical parameters, like the angle and distances between the constituents of the lattice. This allowed them to link the elasticity of the material with the geometry of the lattice. They showed that, contrary to what was thought before, the isotropy was relatively independent of the lattice geometry.

“The conventional wisdom for obtaining isotropy restricts the scope of many applications such as space-filling in 3D-printing,” says Prof Naskar.

^{Left: Macroscopic view of forces or deformation that materials are subjected to. Middle: Microscopic view of lattice structure of a material with one component. Right: Microscopic view of lattice with multiple components.}

^{[Image Credits: Anisotropy tailoring in geometrically isotropic multi-material lattices by Tanmoy Mukhopadhyay, Susmita Naskar, and Sondipon Adhikari]}

Further, the researchers explored how introducing more materials into the lattice changes the isotropy. They showed that, by increasing the components that make up the lattices but keeping its geometric structure intact, the possibility of obtaining isotropy in three-dimensional structures could be improved more than what was thought earlier. Their study will now allow scientists to achieve the exact kind of anisotropy of the material, by designing the geometric properties as well as constituents of the lattice in specific manners.

“Anisotropy is an important design specification for various systems where different elastic properties are necessary along different directions,” says Prof Naskar. “Control over the structure of the lattice and the three-dimensional mechanical properties are crucial for 3D printing. It is necessary to have sufficient options for lattice configurations from which a designer can choose the most suitable one based on various demands,” adds Prof Tanmoy Mukhopadhyay of IIT Kanpur, another researcher involved in the study.

“The requirements come along with geometrical design and manufacturing constraints,” says Prof Sondipon Adhikari of Swansea University, another author of the study. Understanding the origins of the anisotropy in the very lattice structure of materials enables them to eliminate the constraints. The researchers add that the manufacturing of multi-material lattices has become significantly easier in recent years. Their theoretical study lays out the framework where these advancements can be used for quick application to cases that require a specific amount of isotropy or anisotropy in the material properties. The researchers envisage that such artificially engineered materials would find applications in various fields, including the aerospace industry, soft robotics and biomedical devices and implants.

*This article has been run past the researchers, whose work is covered, to ensure accuracy.*