3D printing puts theoretical mathematics into practice to produce hyperelastic metamaterials

3D printing puts theoretical mathematics into practice to produce hyperelastic metamaterials

Researchers in Italy and Poland demonstrate a mathematical approach to predicting the structure of 3D printed elastic structures. By practically realizing elastic theory, the study opens up the potential of 3D printed metamaterials that act “against their nature”, i.e. a solid plastic that resists compression from one direction, but can be stretched in another.

The unusual properties of these materials are useful in areas of civil and biomechanical engineering as they enable new ways to design and reimagine objects. Examples of “exotic” materials given in this study are woven fabrics, and those with an ability to mimic bone tissue, as in previously reported research at Northewestern University.

The metamaterial behaviour of a 3D printed lock, developed by the German Hasso Plattner Institut

A closer look at the elasticity of lattices

The specific focus of this European study is on an ability to mathematically predict pantographic structures. As in the moving image above, pantographic structures are composed of interlinking parallelograms to make a lattice.


Image demonstrating the pantographic structure of a lattice. Figure via Turco, Golaszewski, Giorgio and Placidi

The structure was originally devised as a mathematical example of metamaterial behaviour, impossible to create with traditional manufacturing. In the study, authors investigate how a particular measure of elastic strain, known as the “Hencky-type model”, can now be realised using 3D printing.

Putting theory into practice

To measure how the materials perform in comparison to numerical predictions, the researchers 3D print two pantographic structures: one orthogonal (including right-angles) and one non-orthogonal (with more obtusely angled parallelograms).

These materials then undergo a tensile test to measure mechanical deformation.


Tensile test on an orthogonal lattice (with right angles) Photo via Turco, Golaszewski, Giorgio and Placidi


Tensile test on a non-orthogonal lattice (obtuse angled) Photo via Turco, Golaszewski, Giorgio and Placidi

Research finds that the numerical simulations of these structures are able to predict the same stresses as the physical lattices. The conclusions therefore reveal that,

It deserves to be investigated how to improve the design of 3D printed fabrics in order to fully exploit the exotic behaviour of higher gradient metamaterials. We remark that the behaviour of higher gradient continua shows many peculiarities which deserve a deeper experimental investigation.

Closing the gap between math and technology

As explained in the abstract of the study, “Current research in metamaterials design is pushing to fill the gap between mathematical modeling and technological applications.” Other examples of closing this gap with 3D printing can be seen in MIT’s research into 3D printed graphene, the optical theory behind instilling color into 3D printed objects, and algorithms to mimic the natural strength of oyster shells.

A full version of Can a Hencky-Type Model Predict the Mechanical Behaviour of Pantographic Lattices? is published online in Advanced Structured Materials journal Volume 69. It is co-authored by Emilio Turco from Italy’s Department of Architecture, Design and Urban Planning; Maciej Golaszewski of the Warsaw University of Technology; Ivan Giorgio from Università dell’Aquila; and Luca Placidi from the International Telematic University Uninettuno.

Featured images shows a demonstration of elastic metamaterial properties in Carbon’s 3D printed elastomeric polyurethane. Photo via Carbon on YouTube

Source 3dprintingindustry.com


Very informative article, i must say. combining 3D printing with science, mathematics , robotics and art has resulted in great output and innovations. I work for a 3D printing service company and i can practically say that 3D printing has grown more flexible and advanced with time. We have so many materials and technologies to choose from. With increasing awareness about this technology has opened gates for 3D printing in all major industries.

Custom Prototypes

Leave a Comment

Comment (required)

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

Name (required)
Email (required)