Phase transitions of crystalline solids have inspired researchers to study a two-dimensional (2D) bistable metamaterial while it transforms from a low- to a high-energy configuration. With the help of theory, simulations, and a macroscopic mechanical model, the scientists from ETH Zürich, Harvard University, and Le Mans University are now able to guide and control the motion of the transformations that take place between the two equilibrium states of a 2D metamaterial. This is of great interest for applications such as soft robotics and four-dimensional printing. Additionally, the research team has introduced new criteria for choosing geometries and structures for 2D auxetics, resulting in better performance and functionality.

Two-dimensional bistable auxetic metamaterials are materials that can go through fully reversible shape and size transformations between with two equilibrium states, like an umbrella that opens, closes and is stable and functional in both cases. The transition is governed primarily by the material’s engineered architecture, and to a lesser extent by their chemical composition.

So far, most of the work in the field has focused on geometry redesign and studies of the materials at the equilibrium states, the equivalent of looking at a domino, only when the tiles are up or when they have already fallen down. “Previous work did not take into account the time-dependent, dynamic behavior of the material, like motion, deformations, or waves. The results of our work have revealed…new details for the dynamics of the wave front that propagates in two dimensions in a auxetic metamaterial sheet,” says Ahmad Rafsanjani, an engineer at ETH Zürich, who has collaborated with the Bertoldi group in Harvard University to build a macroscopic mechanical model with the material. 

To understand the model’s behavior the team looked at similar phenomenon at the microscopic level, in particular phase transformation phenomenon in solids and got inspired to build a phase-field model for this type of metamaterials. ‘’In a way, these metamaterials exhibit a structural analogous to a microscopic phase transformation phenomenon in solids. Each unit cell of a 2D metamaterial represents a crystal at the microscopic level,’’  says Romik Khajehtourian, a postdoctoral scholar at the Denis Kochmann group in ETH Zürich, who worked for the development of a theoretical continuum model for the structure.

Experimental observations of transition waves propagating through two-dimensional (2D) structures; each transition wave is triggered in the fully open configuration by a point load applied by the black indenter. Examples show free transition wave propagation in two directions, wave pinning, wave deflection, and rotation. Credit: PNAS, Kochmann Lab, ETH Zürich

“For example, looking at the iron phase diagram, iron changes its solid-state from BCC (body centered cubic) to FCC (face-centered cubic) when the temperature increases. In our 2D metamaterial, each unit-cell changes its stable states, from open to closed, and the domain wall propagates, similar to phase transformation in iron. Although we got inspired by microscopic models to formulate our macroscopic structure, our model is not a one-to-one analogous to any microscopic system, but only a potential structure,’’ says Khajehtourian. 

The unit cells of the structure were cut with laser out of a thermoplastic acetal homopolymer resin sheet, with a thickness of 1.59 mm. Each cell is composed of four triangular building blocks, attached to each other with slender joints, which allow expansion of the structure upon loading. The unit cells are periodically connected to each other in such a way that they form a homogenous flat network. The system is characterized by two stable static configurations: the unstrained ground state and a volumetrically expanded, equilibrium state of higher strain energy. The researchers also inserted stiff point defects in areas in the structure where the unit cell is locked in one configuration and does not open and close and are similar to precipitates in a crystal. These were made by filling the gaps of an open unit cell with acrylic sheets. 

The researchers poke the metamaterial’s layer to initiate transition waves that propagate through the material with a constant-speed transition front, which transforms the unit cells from the open to the closed state as the domain wall passes by. 

To guide the wave propagation the researchers used the defects, which helped them not only redirect or pin the transition waves and control their velocity, but also to split, delay, or merge propagating wave fronts. “In materials usually we do not have control on defects, which are unwanted most of the time. Here we deliberately put the defects in our structure,” Rafsanjani says. 

“We can see these defects as obstacles, as a rock in the middle of a river, that can change the direction of a wave to go around it or even stop the wave if it is big enough,” adds Khajehtourian.

Despite its simplicity, the model developed by Khajehtourian and Kochman captured very well the qualitative transition front behavior and the numerical results obtained from the continuum model were in good qualitative agreement with experimental findings,  thus offering predictive support in the design of tailored configurations for guided transition fronts.

“This paper shows for the first time [phase] transition waves in 2D structures,” says Corentin Coulais, a professor in soft matter physics at the University of Amsterdam, who did not participate in the study. “So far, nonlinear waves such as solitons and transition waves in metamaterials have been mostly realized in 1D canonical systems. This is an exciting emerging area because it advances our ability to propagate mechanical signal and mechanical energy in a robust fashion. The predictive power of the continuum model the team introduces and validates [experimentally], as well as their ability to use defects to steer the waves in a controlled fashion, is particularly impressive,” he says.

Pablo Zavattieri, head of the multi-scale mechanics and materials by design laboratory at Purdue University, says “These materials achieve such transition by harnessing local instabilities. In engineering, we have always avoided instabilities since they were perceived as being negative for many structural applications. However, this work demonstrates that if we control them, these instabilities can prove very useful and may lead to excellent emerging properties. This novel behavior can now be exploited for applications such as soft robotics, energy absorption, or actuation.”

Read the abstract in PNAS.