##### ASU Learning Sparks

## Lattice Structure & Design for Strength

Lattice structures effectively accommodate stress while still being efficient to produce. The evolution of accommodating load stress on structures began in the 1800s with Maxwell's stability criterion which identified just how many rods were needed in a truss structure. Today, there are many ways to create a lattice structure with advanced technology that helps us select a lattice design based on its function or design.

If I give you a stick and I ask you to break it, chances are you’re going to bend it and snap it in two. You’re not going to try and stretch the stick till it breaks. But why is it easier to bend and break than stretch and break? It boils down to how stresses develop in the stick when you apply these different forces to them. And the stress from bending is higher than that from stretching for the same amount of load. You can use these concepts of bending and stretching to design lattices that can be used for a wide range of applications like light-weighting and energy absorption.

This story starts in the 1800s during a period of expanding construction and growth in infrastructure, and one of the important needs at the time was developing theories for how to design structures like bridges and roof trusses that could survive the loads being placed on them. James Clerk Maxwell (yes, *that* Maxwell, of electromagnetics fame) published a paper in 1864, which discussed the stability of rigid truss-like structures, which was useful in answering questions like “how many truss rods do I really need in a bridge?”

Maxwell proposed a way to formulate what were the just-right number of rods in a truss structure. Too few and the structure will deform excessively under load, too many and the rods in the structure introduce unnecessary stresses in themselves.

This idea is captured in what is known as Maxwell’s stability criterion and involves the computation of a metric M with b struts and j joints as follows:

- In 2D structures: M = b – 2j + 3
- In 3D structures: M = b – 3j + 6

Per Maxwell’s criterion and assuming the joints are locked, three scenarios are possible:

- If M < 0, the structure is under-constrained – we have too few rods for a stiff structure
- If M = 0, the structure is a rigid framework – this is just right
- If M > 0, the structure is over-constrained – this results in internal stresses – and unnecessary mass

Nearly 150 years later, as additive manufacturing or 3D printing made it possible to now fabricate truss-like structures in 3D printed parts, designers started borrowing ideas from atomic arrangements in metal crystals. In materials science we call the way in which atoms are arranged in crystal a lattice – and hence when it came time to creating 3-dimensional truss-like structures, we also borrowed the term. And since truss rods conjures images of large bridges and rooftops, we used the word strut instead.

The question materials scientists and designers were asking now was however similar to the one civil engineers were asking in the 1800s: how many lattice struts do I really need? And the ideal number for a stiff lattice is one that gives an M value of 0 – these are lattices that transfer loads axially – which we call stretch dominated lattices. However not all lattices need to be stiff. In some applications like the sole of your shoe or the cushioning foam in a bicycle helmet, we would like the lattice to deform and absorb energy. For these applications, we can choose a lattice design that gives us an M value less than 0 – it is under-constrained, and the individual struts deform by bending – this is what we call a bending-dominated design.

There are of course many different ways of designing lattices – and many applications, some of which may even not be structural in nature, where knowing if the lattice is stretching or bending dominated is useful.

Today we design lattices using powerful computational design tools. We select a particular lattice based on its performance in simulation or experiments – or even because it evolved in nature for a particular function that is of interest to us. But every now and then, we whip out Maxwell’s criterion and check how many struts and joints there are, and predict if the lattice will be stiff or compliant under the application of loads.