Controlled Unusual Stiffness of Mechanical Metamaterials

Mechanical metamaterials that are engineered with sub-unit structures present unusual mechanical properties depending on the loading direction. Although they show promise, their practical utility has so far been somewhat limited because, to the best of our knowledge, no study about the potential of mechanical metamaterials made from sophisticatedly tailored sub-unit structures has been made. Here, we present a mechanical metamaterial whose mechanical properties can be systematically designed without changing its chemical composition or weight. We study the mechanical properties of triply periodic bicontinuous structures whose detailed sub-unit structure can be precisely fabricated using various sub-micron fabrication methods. Simulation results show that the effective wave velocity of the structures along with different directions can be designed to introduce the anisotropy of stiffness by changing a volume fraction and aspect ratio. The ratio of Young’s modulus to shear modulus can be increased by up to at least 100, which is a 3500% increase over that of isotropic material (2.8, acrylonitrile butadiene styrene). Furthermore, Poisson’s ratio of the constituent material changes the ratio while Young’s modulus does not influence it. This study presents the promising potential of mechanical metamaterials for versatile industrial and biomedical applications.


Analytical implementation of the deflection of beam
In order to analytically assess the deflection of beam in triply periodic bicontinuous structures under compressive and shear loadings, we employ Euler-Bernoulli beam theory. 1 In the Euler-Bernoulli beam theory, the deflections of beam under compressive and shear loadings are defined to be inversely proportional to the square and the forth power of beam diameter, respectively. When the direction of loading is parallel to the axial direction of beam, the compressive ( ) and shear ( ) deflections are expressed as 4 / and 64 /3 , respectively. is magnitude of applied loading, , are length and diameter of beam, and is Young's modulus of base material, respectively. The cross sectional shape of beam is assumed to be a circle. The theory shows that the deflection under shear loading is more affected by the change of beam diameter compared to that under compressive loading.

Deflection of beam in D and G structures
When loading is applied on D and G structures, the loading are divided into two components, the Then, the deflection of D and G structures can be calculated by summation of and along the direction parallel to the compressive loading, that is 2 / 32 /3 . In the same manner, deflection of D and G structures under shear loading can be calculated as 2 / 32 /3 . Therefore, the ratio of deflections from compressive and shear loading ( / ) becomes consistent regardless of the diameter of beams as follows: The ratio of Young's modulus to shear modulus (E/S) of structure is linearly proportion to the ratio of deflections from compressive and shear loading ( / ). As a result, the E/S of D and G structures can be consistent regardless of the volume fraction.

Deflection of beam in P structure
The connection angle of simplified beams in P structure is 90°. Thus, the magnitudes of and in P structure are changed with respect to the direction of loading. When the compressive loading is As a result, the E/S of P structures is increased as the volume fraction decreases.

Numerical simulation details Effective wave velocities
The effective longitudinal and transverse wave velocities were calculated as a function of geometric variables with respect to the various structures. All calculations was based on the long wavelength condition where the wavelength (λ=100L) is much larger than the size of unit cells. The effective wave velocity was calculated by using v=w/k where k→0. As shown in figure S1, when the volume fraction changes, the effective transverse wave velocities of P, D, and G structures have similar range while the longitudinal wave velocity of P structure is much greater than those of other structures. When the aspect ratio is changed, P structure shows a relatively little variation of the effective longitudinal wave velocity compared to those of D and G structure. and G structures as a function of volume fraction and aspect ratio.

Effect of aspect ratio on dynamic moduli
For more sophisticated analysis of the effect of aspect ratio, we have calculated the elastic wave velocity of the structures along various directions (0 o , 45 o , and 90 o ). Figure S2 shows

Boundary conditions in finite element analysis
In order to calculate Young's modulus and shear modulus, we applied the boundary conditions on the surfaces of the structure as shown in figure S3. We applied fixed boundary condition on the bottom of structure. The lateral surfaces of structures were set to be free. Compressive and shear loadings were applied on the top surface of structures along the z-direction. The magnitude of applied loading was controlled to be less than 1% that satisfied the small strain approximation. Figure S3. Applied boundary conditions in the finite element analysis.