Text from the interactive explanation

Sequential metamaterials with alternating Poisson’s ratios

Amin Farzaneh, Nikhil Pawar, Carlos M. Portela, Jonathan B. Hopkins

A living figure from Farzaneh, Pawar, Portela & Hopkins · Nature Communications 2022

This lattice cell is being stretched and squeezed along a single line. Watch its side tabs. As the stretch keeps growing they move in — then, the instant two small bumpers touch, they snap out . No motor, sensor or battery decides when. The sequence is built into the shape.

Pull 0.0 %

grey trace: the pull

mint trace and arrows: sideways motion (trace drawn 2.5× taller)

amber: a hard stop closing

blue / red: the two beam families

Live simulation of the paper’s Design I cell (Teflon, ±15 % pull), computed in your browser with the paper’s analytical method. The faint shadow is the cell at rest.

Why would anyone want a material that changes its mind? The authors picture machines that coordinate motion without electronics: one actuator whose single back-and-forth push comes out as several synchronized motions of different sizes and rhythms; a strip that ripples forward like a caterpillar from one motor; a mechanical “key” that trips a chain of logic switches in places where electronics fail.

All of that needs one thing first: a material whose sideways response switches direction on schedule. This page walks you through how they built one — and lets you build your own.

1

The sideways rule

Pull a rubber band and it gets skinnier. Engineers describe this with two ideas. Strain is how much something stretches, as a fraction of its length: 10 % strain means 10 cm longer for every metre. Poisson’s ratio (written ν, “nu”) compares the sideways strain to the pulling strain, with a minus sign so that ordinary materials, which get thinner, come out positive. Most solids sit near +0.3. Teflon — the slippery plastic the paper’s large test cells are cut from — is +0.46.

But the inner shape of a material can overrule what it is made of. Here are three swatches, pulled by exactly the same amount.

Predict Before you stretch: will the bow-tie lattice on the right get thinner or fatter?

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Make your prediction above — the answer appears here.

Illustrative sketch: rigid walls joined by hinges (the solid strip uses Teflon’s ν = 0.46). Hinged sketches exaggerate the size of the effect; the point is its sign.

What you just saw

The regular honeycomb, whose blue walls point outward, narrows like the solid. The bow-tie or re-entrant honeycomb, whose red walls point inward, widens: a negative Poisson’s ratio. Materials that do this are called auxetic . Structures whose behaviour comes from their architecture rather than their ingredients are called mechanical metamaterials , and auxetic ones have been proposed for hip implants that load the surrounding bone, bandages whose pores open as a wound swells, and materials that hug an object that hits them.

Every one of them, though, answers a pull in one fixed way. The paper’s question: could a material be designed to switch between the two answers — abruptly, repeatedly and in a chosen order — while it is pulled along one unchanging direction?

2

Two beams, one chain

The cell in the opening scene borrows a piece of each honeycomb. Its thin blue beams face outward, like the regular honeycomb: stretching them drags the corner blocks — and so the sides — in. Its red beams face inward, like the bow-tie: stretching them swings the side arms out. The two families are linked end to end, so when you pull the cell the same force runs through both. Who wins the tug-of-war over the side tabs?

Try Slide toward “red beams softer” and watch the arrows. Where do the sides change their mind?

Simulation: the paper’s cell model with every hard stop switched off (their parts are drawn faded). Beam thicknesses are varied around Design I, keeping their product fixed.

What you just saw

Whichever family is softer bends more under the shared force, and its sideways habit takes over. In the paper’s Design I the blue beams are the flexible ones (2.1 mm thick and long, against 3.8 mm for the red), so the sides move in. Make the red beams soft enough and the red arrow outgrows the blue one: the sides move out. The authors call this first ingredient differential stiffness .

The paper’s analytical method treats the cell as stiff blocks (“rigid bodies”) joined by slender beams that bend according to classical Euler–Bernoulli beam theory. The bottom tab is held fixed; the top tab is nudged up by a tiny amount and a stiffness matrix tells where every block moves. Then the beams are rebuilt straight between the new block positions and the next nudge is applied — hundreds of times per cycle. When two blocks touch at a hard stop, they are treated as fused (or, for the side tab, allowed to slide). The simulations on this page run that same procedure in your browser.

3

A switch made of shape

Now add the second ingredient: hard stops , features designed to collide. The two long inner arms of the cell end in a ball and a socket that start a few millimetres apart. While the soft blue beams do the work, the corners swing inward and that gap closes. When ball meets socket, the blue part can no longer move the sides. Every further bit of stretch has to flow through the red beams — and the sides reverse.

Goal Make the sides turn around at exactly 6 % stretch . Drag the gap.

Simulation: Design I in tension with only Hard stop (1), the ball and socket, switched on. Gap 4.9 mm is the paper’s Design I.

What you just saw

The gap works like a timer whose clock is the stretch itself. A smaller gap closes sooner and the turnaround comes earlier; a gap that is too wide never closes within the pull, and the sides simply keep moving in. Because the arms meet through a curved ball and socket, the two halves can still rotate against each other after contact — which rubs. That is why the authors recommend a low-friction material such as Teflon for designs that rely on this stop.

4

One rhythm in, a faster rhythm out

Real machines push and pull back and forth. So the authors gave Design I one smooth wave of stretch and squeeze. Their simulations spread a single cycle over 60 minutes, slow enough that nothing shakes or bounces (engineers call this quasi-static). Different hard stops guard each half: stop (1) in the stretch; stop (2), then — for big enough squeezes — stop (3).

Count Drag the amplitude from 5 % up to 25 % and count how often the side tabs turn around.

Simulation of Design I (elastic material, cell inside an endless lattice). The optional dashed overlay is the output saved with the authors’ published MATLAB tool for the same design at 15 %.

What you just saw

At a gentle 5 % nothing collides and the cell behaves like an ordinary material: sides in when stretched, out when squeezed — one rhythm, matching the input. Past about 9 % the ball meets its socket and the sides start alternating, so they now change direction more often than the input does. That is what the authors mean by passively multiplying an actuator’s frequency. Push past roughly 17 % and the squeeze also drives the side tabs against the corner pieces — stop (3). From then on the sides barely move while the squeeze carries on: in this simplified model they creep only slowly, and in the paper’s more detailed FEA they hold perfectly still at about −9 % (Fig. 3b).

The dashed purple line is Poisson’s ratio itself, now flipping between positive and negative within each cycle: an alternating Poisson’s ratio , the title of the paper.

5

Write a sequence

Every dimension in the cell’s drawing is a dial. Beam thicknesses decide who bends first; arm lengths and clearances decide when each stop closes. Tune them and you write a new sequence. The authors’ Design II mainly swaps which beams are stiffer, and its sides answer the opposite way (the paper’s Fig. 4).

Simulation with the paper’s analytical method (elastic, quasi-static, cell inside an endless lattice). Dimensions in millimetres for a cell 100 mm across, as in Supplementary Table 1.

Why speed matters

Designing a sequence means trying many shapes, and the standard brute-force simulation, finite element analysis (FEA), carves the part into thousands of small blocks and is slow. On a desktop computer the authors’ analytical tool computed Design I’s full cycle in 21 seconds; the elastic and plastic FEA runs took 1670 and 2002 seconds. Across both test designs the tool was 39.7 to 95.3 times faster — fast enough to design with, which is the advance the authors highlight in their abstract. The same approach finishes a cycle in milliseconds in your browser.

6

Stack them: one pull, four answers

One cell gives one sequence in time. Stack rows of different cells and you also get a pattern in space. The authors designed four cells, A to D, each tuned to answer the same 13 % stretch differently, and laser-cut a Teflon lattice with one design per row.

Compare Make every row the same, then go back to the graded lattice. Watch the lattice’s outline.

Simulation of each row as if it stood alone under 13 % stretch — the “designed” behaviour of the paper’s Fig. 6c. Row geometries from Supplementary Table 2.

What you just saw

With identical rows, the whole lattice breathes in step. In the graded lattice each row follows its own score: rows A and B pull in and then partly recover, while C and D push out. The outline ripples as the pull rises and falls. A Poisson’s ratio that alternates in time and across space is what could one day let a single actuator make a material undulate.

Joined together and measured, each row still answered differently, but not exactly as designed. The paper gives two reasons. Neighbouring rows are physically connected, so they frustrate each other’s attempts to widen or narrow. And the rows differ in stiffness, so the soft rows stretch more than the stiff ones; none actually feels the 13 % it was designed for. The authors’ remedies: separate rows with vertical flexures, and either match the rows’ stiffness or design each row for the strain it will really receive.

7

Does the real thing agree?

The authors checked their tool twice: against FEA (verification) and against real cells (validation). They laser-cut Design I and Design II from sheets of Teflon about 15 cm across, put red dot markers on the side tabs, and filmed them being stretched and squeezed in an Instron testing machine while image-tracking software followed the dots.

For Design I, the analytical tool, an elastic FEA, a plastic FEA (which also lets the Teflon bend permanently and includes gravity) and the measurements all trace the same alternating curve. For Design II the 15 % strain was large enough to make the cell yield — bend permanently — which the tool does not model, so tool and measurement part ways after the peak stretch. The lesson the authors draw: as long as the cells do not appreciably yield, and are made and clamped carefully, the tool predicts their alternating Poisson’s ratios accurately.

Race Computer time to predict Design I’s cycle (paper, Fig. 5a), replayed to scale.

computer time elapsed: 0 s

From hand-sized to microscopic

The idea is not limited to flat sheets. Copy a flat cell, rotate the copy 90° about its vertical axis and merge the two, and you get a 3D cell with the same alternating behaviour and twice the stiffness — once the ball becomes a sphere and some flat stops become wedges.

Using two-photon lithography — a tightly focused laser that hardens liquid resin only at its focal point, drawing 3D structures point by point — the team printed such a cell about a tenth of a millimetre tall, with a custom grip on top. A nanoindenter (a tiny mechanical tester) then stretched and squeezed it twice inside a scanning electron microscope.

The micro-cell’s sideways strain alternates as designed, cycle after cycle, with a little hysteresis: the second cycle differs slightly from the first because the material yielded a bit. A 3 × 3 × 3 micro-lattice, tested only in compression, showed the alternation in every row, but friction at the tester’s tip and small printing imperfections made it buckle early. The authors expect larger, more precisely printed lattices to behave more like their single cells.

8

Back to the pull

Before this work, materials could have negative or positive Poisson’s ratios, or ratios that drift gradually as they deform. None could abruptly reverse the direction of their sideways motion — back and forth, in a designed order, and differently from place to place — while being loaded along one unchanging direction.

The paper shows how: soft-before-stiff beams, plus collisions timed by millimetres of clearance. It supplies a design tool fast enough to use, and demonstrates the effect in hand-sized Teflon cells, a four-row graded lattice and 3D cells printed a tenth of a millimetre tall.

What it brings closer are the applications the authors propose, still to be built:

One actuator, many outputs. Several synchronized motions with coupled but different sizes and rhythms from a single input — for example doubling its frequency while refining its step size.

Single-actuator locomotion. Rows that alternate in a choreographed order could make a material undulate like a walking caterpillar.

Mechanical keys. A material that delivers just the right sequence of pushes to a bank of bistable mechanical logic gates, for simple but critical computations where electronics are likely to fail.

In this material the sequence is not stored on a chip. It lives in the thickness of a beam and the width of a gap.

Pull the cell again ↑

Source & credits

Amin Farzaneh, Nikhil Pawar, Carlos M. Portela & Jonathan B. Hopkins. Sequential metamaterials with alternating Poisson’s ratios. Nature Communications 13, 1041 (2022). https://doi.org/10.1038/s41467-022-28696-9

The article and its supplementary materials are open access under a Creative Commons Attribution 4.0 International License.

Figures 1, 3, 5, 6, 7 and 8 are reproduced from the article; some are resized and converted to WebP.

Supplementary Movies 4, 6 and 7 are shown as excerpts: trimmed, resized, re-encoded to WebM and with audio removed.

The simulations on this page are a JavaScript re-implementation, adapted for the browser, of the method in the authors’ MATLAB tool (Supplementary Software 1, © 2022 Flexible Research Group @ UCLA, Apache License 2.0). For Design I at 15 % it reproduces the tool’s saved output to within about half a percent strain. Design dimensions come from Supplementary Tables 1 and 2.

The honeycomb swatches in chapter 1 are illustrative hinged-wall sketches, not results from the paper.

What is what: the cells on this page are simulated (elastic material, slow loading, cell inside an endless lattice); the figures and videos show the paper’s FEA simulations and measured experiments; the applications in chapter 8 are proposed by the authors.