Most materials—from rubber bands to steel beams—thin out as they are
stretched, but engineers can use origami's interlocking ridges and precise
folds to reverse this tendency and build devices that grow wider as they are
pulled apart.
Researchers increasingly use this kind of technique, drawn from the ancient
art of origami, to design spacecraft components, medical robots and antenna
arrays. However, much of the work has progressed via instinct and trial and
error. Now, researchers from Princeton Engineering and Georgia Tech have
developed a general formula that analyzes how structures can be configured
to thin, remain unaffected, or thicken as they are stretched, pushed or
bent.
Kon-Well Wang, a professor of mechanical engineering at the University of
Michigan who was not involved in the research, called the work "elegant and
extremely intriguing."
Wang, the Stephen P. Timoshenko Collegiate Professor of Mechanical
Engineering, said the paper "creates new tools and paths for the technical
community to harness and pursue that will further elevate the
functionalities of advanced origami and metamaterials. The impact is
tremendous."
In a paper published Aug. 3 in the Proceedings of the National Academy of
Sciences, Paulino and his colleagues lay out their general rule for the way
a broad class of origami responds to stress. The rule applies to origami
formed from parallelograms (such as a square, rhombus or rectangle) made of
thin material. In their article, the researchers use origami to explore how
structures respond to certain kinds of mechanical stress—for example, how a
rectangular sponge swells in a bowtie shape when squeezed in the middle of
its long sides. Of particular interest was how materials behave when
stretched, like a stick of chewing gum that thins as it is pulled at both
ends. The ratio of compression along one axis with stretching along the
other is called the Poisson ratio.
"Most materials have a positive Poisson ratio. If, for example, you pick up
a rubber band and stretch it, it will become thinner and thinner before it
breaks," said Glaucio Paulino, the Margareta Engman Augustine Professor of
Engineering at Princeton. "Cork has a zero Poisson ratio, and that is the
only reason you can put the cork back in a wine bottle. Otherwise, you would
break the bottle."
The researchers were able to write a set of equations to predict how
origami-inspired structures will behave under this kind of stress. They then
used the equations to create origami structures with a negative Poisson
ratio—origami structures that grew wide instead of narrower when their ends
were pulled, or structures that snapped into dome shapes when bent instead
of sagging into a saddle shape.
"With origami you can do this," said Paulino, who is a professor of civil
and environmental engineering and the Princeton materials institute. "It's
an amazing effect of geometry."
James McInerney, who is the study's first author and a postdoctoral
researcher at the University of Michigan, said the team created the
equations to understand the property of symmetry in the structures. Symmetry
means something that remains the same under certain transformation. For
example, if you spin a square 180 degrees around an axis running between the
centers of two sides, its shape remains the same.
"Things that are symmetrical deform in expected ways in certain conditions,"
McInerney said. By finding those symmetries in the origami, the researchers
were able to create a system of equations that governed how the structure
would respond to stress.
McInerney said that the process was more complex than defining the symmetry
rules because some of the folds resulted in deformations that did not obey
the rules. He said generally the deformations made in the same plane as the
paper (or thin material being folded) obeyed the rules, and those out of the
plane broke the rules. "They broke the symmetry, but they broke the symmetry
in a way that we could predict," he said.
Zeb Rocklin, an assistant physics professor at the Georgia Tech School of
Physics and a co-author, said that origami presented a fascinating and
contradictory behavior.
"Usually, if you take a thin sheet or slab and you pull on it, it will
retract in the middle. If you take the same sheet and bend it upwards, it
will usually form a Pringle—or saddle—shape. Some materials instead thicken
when you pull on them, and those always form domes rather than saddles. The
amount of thinning always predicts the amount of bending" he said. "The
bending of these origami is exactly the opposite of all conventional
materials. Why is that?"
Researchers have spent years seeking to define rules governing different
classes of origami, with different folding patterns and shapes. But Rocklin
said the research team discovered the class of origami was not important. It
was the way the folds interacted that was key. To understand why origami
seemed to defy movement usually defined by Poisson's ratio—growing wider
when pulled, for example—the researchers needed to understand how the
interaction affected the movement of the entire structure. When artists fold
the sheet so that it moves along its plane—for example, corrugating it so it
can expand and contract—they also introduce a bend that moves the sheet into
the saddle shape.
"It is a hidden mode that comes along for the ride," Rocklin said.
Rocklin said by examining this hidden connection, the researchers were able
to explain "this weird mode of the sheet doing opposite of what was
expected."
"And we have a symmetry of that that explains why it does the exact
opposite," he said.
In the future, the researchers intend to build on their work by examining
more complex systems.
"We would like to try to validate this for different patterns, different
configurations; to make sense of the theory and validate it," Paulino said.
"For example, we need to investigate patterns such as the blockfold pattern,
which is quite intriguing."
The research article, "Discrete symmetries control geometric mechanics in
parallelogram-based origami," was published online Aug. 3 in the Proceedings
of the National Academy of Sciences.
Reference:
James McInerney et al, Discrete symmetries control geometric mechanics in
parallelogram-based origami, Proceedings of the National Academy of Sciences
(2022).
DOI: 10.1073/pnas.2202777119
Tags:
Physics