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Normally, when you stretch any normal material it will get thinner and longer. The negative ratio of strain in the direction of stretching to the perpendicular direction is called the Poisson's ratio \(\nu\). \(\nu\) is typically a positive value.
Auxetic materials however have a negative Poisson's ratio, meaning they expand in the direction perpendicular to the applied stress when stretched. But how is this possible?
One way to achieve this behavior is to design the material with a specific microstructure that allows for this counterintuitive deformation.
In this article we will design such a material, simulate it using FEM (NGSolve) and then 3D print it out of TPU.
There are many possible ways to design such a structure. We will use a rather simple model consisting of a periodic arrangement of re-entrant structures [1]. Looking at a single cell, we can already see how the middle arms will straighten under load, pushing them apart. For a structure like this, this hinge-like rotation of the internal arms dominates compared to pure material stretching causing it to expand.
To understand the mechanism, let's now look at such a unit cell:
Next, we can pattern this unit cell to create a larger structure. After extruding it into 3D we end up with fig. 2.
To simulate the behavior of the meta material, we will use the finite element method (FEM) with NGSolve. The full code for this example can be found on the NGSolve usermeeting page. We will not go into detail here but provide a brief overview.
First we need to mesh the geometry. In this case, we have already created a geometry that will mesh nicely. For the actual 3D print, the outside of corners will get a chamfered edge with radius equal to the nozzle radius. This is to improve the print quality as the printer cannot reproduce sharp outside corners. We do not include these chamfers in the simulation as it would require small elements to resolve the tiny radii (even when curving the mesh).
In this example we end up with \(36282\) elements.
For this example we will use a linear elasticity model, and invert the matrix directly.
As boundary conditions we will fix the left side of the model and apply a force in \(X\) direction on the right side. The resulting deformation is shown in fig. 3.
For larger deformations we see the limitations of this approach. The material starts to bend outwards which is clearly unrealistic. This happens because linear elasticity does not account for large deformations and geometric nonlinearities.
For small deformations however, we see the expected "negative Poisson's ratio" behavior. The material expands in the direction perpendicular to the applied force when stretched.
As the visualization is interactive, the "Open Controls" in the top right corner can be used to change the deformation amount.
Finally, we can 3D print the model and test it out in real life. We will use TPU for this as it is a flexible material that can stretch and show the auxetic behavior.
The model was somewhat optimized for 3D printing. As mentioned, outside corners are chamfered to improve tool path generation. Furthermore, the wall thickness was chosen to be \(2x\) the nozzle diameter. This way the printer can print each cell as a closed loop, reducing retractions and oozing, issues we want to avoid when printing soft tpu.
Models with thinner walls were also printed from PLA but they did not work that well. The printed poorly and were too stiff.
The video below shows the printed model in action. It can be stretched and behaves like we have already seen in the simulation.
Auxetic materials have a wide range of potential applications due to their unique properties. They can be used in areas such as:
Overall I really like this project. It covers the design, simulation, and manufacturing of such an auxetic material. The result is a really nice show piece. While this project is intentionally kept simple, let's discuss its limitations for completeness.
One limitation of the simple design used is that it's Poisson's ratio is anisotropic, meaning it varies with direction. If we stretch it diagonally for example, it behaves like a conventional material.
More complex designs from e.g. topology optimization can help to alleviate this issue.
The simulation we used is simple, fast, and shows the behavior of the auxetic material under load. However, as discussed, there are limitations to this approach.
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