Overview

The adidas shoe "4DFWD," which has a special 3D-printed structure embedded in the midsole, has the function of converting stepping force into propulsive force. This article interprets this function as a mechanical metamaterial that converts compression into shear and explains how to design it.

This article explains the first of the following three parts.

(1) Explanation of macro physical properties that convert compression to shear

(2) Design of mechanical metamaterials that convert compression to shear

(3) Designing a midsole that converts stepping force into propulsive force

Material properties that convert compressive stress into shear strain \(S_{32}\).

In the previous article, we explained the material property that converts compressive stress into shear strain in terms of a two-dimensional linear elastic constitutive law, which is the \((3,2)\) component \(S(\theta)_{32}\) of the compliance matrix \(\mathbf{S}(\theta)\). We have clarified that This value was expressed using the material rotation angle \(\theta\) as follows

\(S(\theta)_{32} = \sin{4\theta}\cdot\frac{1}{4}\left(\frac{1}{G} - \frac{2(1+\nu)}{E}\right) = S(\theta)_{ang} \times S_{mat}\tag{1-1}\)

where \(S(\theta)_{32}\) has been separated into a term \(S(\theta)_{ang}\) for the rotation angle and \(S_{mat}\) for the material properties.

\(S(\theta)_{ang} = \sin{4\theta}, \quad S_{mat} = \frac{1}{4}\left(\frac{1}{G} - \frac{2(1+\nu)}{E}\right)\tag{1-2}\)

Therefore, the maximization of \(S(\theta)_{32}\) can be done independently of the rotation angle and material properties.

Maximization of \(S(\theta)_{ang}\).

Maximizing \(S(\theta)_{ang}\) is easy. It takes the maximum value \(1\) when the rotation angle \(\theta\) is \(22.5^\circ\). Therefore, \(S(\theta)_{32}=S_{mat}\) at this time.

Maximizing \(S_{mat}\).

When the material is isotropic, this value is 0. To increase the absolute value of this value, the ratio of Young's modulus \(E\) to shear modulus \(G\) must be large or small. In other words

\(E>>G, \quad \text{or} \quad E<
then compressive stress is converted to shear strain to a large extent.
So what unit cells with such material properties are possible?

The following two examples are presented next.

X-lattice with low Young's modulus \(E\) and high shear modulus \(G\).
+Lattice with high Young's modulus \(E\) and low shear modulus \(G\).

Homogenization of unit cells
Before introducing each lattice, we first introduce the [homogenization method](https://www.jstage.jst.go.jp/article/seikeikakou/29/2/29_42 /_article/-char/en/)[1][2], which is a method for calculating the equivalent compliance matrix S$ when lattice is considered a homogeneous material.

The homogenization method takes a unit cell of Lattice, a periodic structure, as shown below, and considers it as a homogeneous material.

Its macrocompliance matrix \(\mathbf{S}_{macro}\) gives the macrostresses \(\sigma_x, \sigma_y, \gamma_{xy}\) for each unit cell as shown below, and

The corresponding macrostrain \(\boldsymbol\varepsilon_{\sigma_x}, \boldsymbol\varepsilon_{\sigma_y}, \boldsymbol\varepsilon_{\gamma_xy}\) is calculated.

The boundary surfaces of the unit cells are given periodic boundary conditions to constrain them to deform in a periodic manner.

The macrocompliance matrix is then obtained by arranging them as follows.
\(\mathbf{S}_{macro}=
\left[
\frac{\boldsymbol\varepsilon_{\sigma_x}}{\sigma_x} \quad
\frac{\boldsymbol\varepsilon_{\sigma_y}}{\sigma_y} \quad
\frac{\boldsymbol\varepsilon_{\gamma_{xy}}}{\gamma_{xy}}
\right]
\tag{2-1}\)
From the macrocompliance matrix thus obtained, we obtain the macro Young's modulus \(E_{macro}\), the macro shear modulus \(G_{macro}\), and the macro Poisson's ratio \(Nu_{macro}\).

In this case, since the geometry has mirror symmetry in the XZ and YZ planes and 90° rotational symmetry, the macro linear elastic material is classified in the class of square anisotropy, and can be obtained as follows.
\(E_{macro}=\frac{1}{S_{macro,11}},\quad G_{macro}=\frac{1}{S_{macro,33}},\quad \nu_{macro}=-\frac{S_{macro,12}}{S_{macro,11}}
\tag{2-2}\)

X lattice with low Young's modulus \(E\) and high shear modulus \(G\).
Now we will measure the macro property values of specific lattices using the homogenization method.

A lattice structure such as the X-lattice shown in the figure above (hereafter referred to as X-lattice) is

Deforms by bending beams against tensile and compressive forces
Shear forces are resisted by the axial force of the beams.

Therefore, the material properties of the unit cell are

Macroscopic Young's modulus \(E_{macro}\): low
Macro shear modulus \(G_{macro}\): High

The material properties of the unit cell are expected to be as follows.
Carbon's resin for optical modeling EPU40

Young's modulus: \(8\text{[MPa]}\).

Poisson's ratio: 0.5

Assuming

Assuming that the unit size is \(10/\sqrt{2}\text{[mm]}\) and the plate thickness is \(1\text{[mm]}\), the macro material properties of this unit cell were actually measured using the homogenization method, and the following results were obtained.

\(E_{macro}=0.210\text{[MPa]}\)
\(G_{macro}=0.904\text{[MPa]}\)
\(\nu_{macro}=0.890\)

The result is as follows. As expected, the macro shear modulus is higher relative to the macro Young's modulus. Interestingly, the macro-Poisson's ratio was about 0.9, which is very high compared to the normal material. \(S_{mat}\) follows from the expression \(\text{(1-2)}\).
\(S_{mat} = -4.22\text{[1/MPa]} \tag{3-1}\)
For example, if we rotate by \(\theta=22.5°\), we get
\(S(\theta)_{32}=1.0\times(-4.22)=-4.22 \tag{3-2}\)
and from the previous article equation \(\text{(2-1)}\) we see that the compressive force is converted to a positively oriented shear deformation as follows.
\(\gamma_{xy} = C'_{32}\times(-\sigma_y)=4.22\sigma_y \tag{3-3}\)
Young's ratio \(E\) is high and shear ratio \(G\) is low + Lattice

A lattice structure such as the one shown above with + laid on top (hereafter referred to as + lattice structure) is

Axial force of the beam against tensile and compressive forces
The + lattice structure is deformed by the bending of the beams against shear forces.

Therefore, the material properties of the unit cell are

Macroscopic Young's modulus \(E_{macro}\): low
Macro shear modulus \(G_{macro}\): High

and is expected to be \(G_{macro}\): high.
Similarly, when EPU40 is used

Similarly, when the unit size is \(5\text{[mm]}\) and the plate thickness is \(1\text{[mm]}\), the macro material properties of this unit cell were actually measured using the homogenization method, and the following results were obtained.

\(E_{macro}=1.854\text{[MPa]}\)
\(G_{macro}=0.055\text{[MPa]}\)
\(\nu_{macro}=0.027\)

The result is as follows. As expected, the macro shear modulus is low relative to the macro Young's modulus. Interestingly, the macro-Poisson's ratio was about zero, a low value similar to that of cork material. \(S_{mat}\) follows from the equation \(\text{(1-2)}\).
\(S_{mat} = 4.29\text{[1/MPa]} \tag{3-4}\)
For example, if we rotate by \(\theta=22.5°\), we get
\(S(\theta)_{32}=1.0\times(4.29)=4.29 \tag{3-5}\)
and from the previous article equation \(\text{(2-1)}\) we see that the compressive force is converted to a negatively oriented shear deformation as follows.
\(\gamma_{xy} = S(\theta)_{32}\times(-\sigma_y)=-4.29\sigma_y \tag{3-6}\)
Comparing with the expression \(\text{(3-3)}\), we can see that the signs are reversed and the magnitudes are almost the same.

Therefore, we can infer that there is some relationship between X-lattice and +-lattice.
Relationship between X-Lattice and +-Lattice
The two lattices presented above actually overlap exactly when rotated by 45°, which means they are the same lattice.

Naturally, the macro-compliance matrices are also nearly identical when rotated by 45°.
Also, the X and + lattices, each rotated 22.5°, are mirror-symmetric, as shown in the figure below.

Thus, this geometric symmetry is the reason why the compression-shear components \(S(\theta)_{32}\) of the X-lattice and +-lattice are of the same magnitude with sign reversal from the expressions \(\text{(3-2)}, \text{(3-5)}\) respectively.
Verify deformation characteristics by numerical simulation
Setup
Numerical simulation is used to verify that the compressive force is actually converted into shear deformation.

The geometry is an X-lattice, rigid blocks are connected to the top and bottom surfaces, the bottom surface is fixed, and a downward force of 500 N is applied to the top surface to measure the lateral displacement \(u_x\) of the top surface. The analysis method is a simple linear static analysis.

The rotation angles are \(\theta = 0^\circ, 11.25^\circ, 22.5^\circ, 33.75^\circ, 45^\circ\).

When \(\theta=45^\circ\), it has the same shape as +Lattice.
Result.
When \(\theta=0^\circ,45^\circ\), \(u_x=0\text{[mm]}\) and \(\theta=11.25^\circ,22.5^\circ,33.75^\circ\), the lateral displacement occurs and as calculated \(\theta=22.5^\ circ\), the maximum value \(u_x=3.74\text{[mm]}\) is obtained. From this we know that we were able to design a mechanical metamaterial in which compressive forces are converted into shear deformation.

Prototype using FDM 3D printer
Prototyping
Finally, let's conduct an experiment with an actual prototype to see if it behaves as designed.

The test piece will be fabricated using an FDM (thermal melting and stacking) 3D printer.

The material used is TPU, which is a soft material, because we want to deform the specimen significantly.

The compression test is performed using a load displacement tester.
Result

The result is shown in the figure above.

As in the simulation, the largest shear deformation occurred at \(\theta=22.5^\circ\).

The deformation is stable at \(\theta=0^\circ,11.25^\circ,22.5^\circ\), while

\theta=33.75^\circ, 45^\circ$ buckled due to compressive forces.
Compression test video



The video of the compression test clearly shows that shear deformation is induced with and without rotation.

The top surface slides significantly during shear deformation, a deformation that seems to create propulsive force.
Summary
From the viewpoint of material mechanics, it was clarified that in order to convert compressive force into shear deformation, a special material with a large Young's modulus to shear modulus ratio can be rotated. The above properties were confirmed by measuring equivalent macroscopic material properties using the homogenization method. We also understood from the geometric and material mechanics perspectives that the X and + lattices coincide when they are rotated. Finally, through simulation and prototype experiments, we confirmed that we were able to design a mechanical metamaterial in which compressive forces are converted to shear deformation.
In the next article, we will design a midsole that converts stepping force into propulsive force using the rotated X-lattice.
References
[1] K, Terada, Analysis of composite materials using homogenization method, Molding and Processing, 2017, Vol. 29, No. 2, p. 42-46, Published on Sep 10, 2020, Online ISSN 1883-7417, Print ISSN 0915-4027
[2] K. Terada, N. Hirayama, K. Yamamoto, Numerical material testing: Multiscale analysis by finite element method, Maruzen Publishing, 2021, ISBN 978-4-621-30633-8