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Triply periodic bicontinuous structures were defined by a volume enclosed by triply periodic minimal surfaces^{20,21,22}. The representative triply periodic bicontinuous structures, including Schwarz’ primitive (*P*), diamond (*D*) and Schoen’s gyroid (*G*), are illustrated in Fig. 1a. Literature describes the *P*, *D* and *G* structures by using the level surfaces of the following trigonometric functions^{23,24,25}:

where *t* and α stand for the threshold of level surface and aspect ratio, respectively. The threshold determines a volume fraction (φ) that is defined by the ratio of the volume enclosed by the level surface to the volume occupied by the structure. The structures consist of basis atoms and connections between basis atoms (Fig. 1b). The aspect ratio is defined by the ratio of the lattice constant along the tensile to shear direction. We choose the volume fraction (φ) and aspect ratio (α) as geometric variables that characterize the geometry of structures. Figure 1c shows the various geometries of *P* structures with respect to different volume fractions (20%, 40% and 60%). When the structure is fabricated by a photolithography process, the volume fraction of the structure defined by trigonometric functions is determined by the exposure time and intensity of the laser. During the crosslinking reaction of negative photoresist (SU8), the cross-linked part of the photoresist can be reduced as the exposure time and intensity of the laser decrease. As the exposure time and intensity of the laser decrease, the volume fraction of the structure also decreases and the structure can collapse due to the diminished connections between the basis atoms (pinch-off)^{17}. The volume fractions of *P* and *D* structures are about 20% and that of *G* structure is about 5% at pinch-off ^{26}. Figure 1d shows the geometry of *P* structures with respect to the different aspect ratios (1, 1/2 and 1/3). When the aspect ratio is unity, the structure has an identical shape along the *x*, *y* and *z*-directions. By changing the aspect ratio, the structure becomes a transverse isotropic structure.

In elastic mechanics, the stiffness of a structure can be characterized by elastic wave velocities. To investigate the stiffness of the triply periodic bicontinuous structures (*i.e*., *P*, *D* and *G*) with the propagation of elastic waves, we have calculated the effective wave velocity of each structure under long wavelength condition. The effective longitudinal and transverse wave velocities of *P*, *D* and *G* structures in different directions are shown in Fig. 2a–c, where the 0^{o} and 90^{o} represent *x* and *z*-directions, respectively. Simulations show that the effective wave velocity of *P* structure has much greater variation along with different directions than those of *D* and *G* structures. Specifically, in the 0^{o} and 90^{o} of directions of *P* structure, the longitudinal wave velocity is about 400% greater than the transverse wave velocity. It indicates that the *P* structure is the most anisotropic structure among *P*, *D* and *G* structures. Then, we have retrieved the effective elastic parameters. The triply periodic bicontinuous structures are fabricated by the interference lithography and only the direction of 90^{o} with respect to a substrate is practically allowable in applications. Thus, we calculated the ratio of longitudinal modulus (*M*_{eff}) to shear modulus (μ_{eff}) only in the direction of 90^{o} to compare the behaviors of effective elastic parameters according to the changes of geometric variables. As shown in Fig. 2d, the *M*_{eff}/μ_{eff} ratio of *P* structure is significantly increased as the volume fraction of structure decreases, while those of *D* and *G* structures show slight changes. We also investigated the effect of aspect ratio on the *M*_{eff}/μ_{eff} ratios of structures (Fig. 2e). It clearly reveals that the *P* structure have wider attainable range of *M*_{eff}/μ_{eff} ratio than others. Interestingly, the *M*_{eff}/μ_{eff} ratio of *P* structure is increased drastically as the aspect ratio decreases below the unity. It is caused by the distinctive trend of longitudinal wave velocity of *P* structure with respect to the change of aspect ratio (see Supplementary Information).

In order to study the stiffness of structures with their elastic moduli, the ratios of Young’s modulus to shear modulus in the direction of 90^{o} according to the changes of geometric variables are investigated. First, the ratios of Young’s modulus to shear modulus (*E*/*S* ratio) of *P*, *D* and *G* structures as functions of volume fraction are calculated and are shown in Fig. 3a. *E*/*S* ratios of *D* and *G* structures does not change from around 4.4 and 3.7 even though the volume fractions of *D* and *G* structures increase considerably from 20% to 80% and from 5% to 80%, respectively. Interestingly, *E*/*S* ratio of *P* structure increases significantly as the volume fraction decreases from 80% to 20%. When the volume fraction of *P* structure becomes about 20%, *E*/*S* ratio increases to 12, which is about 300% larger than that of isotropic bulk material, which is 2.8. In this case, the Young’s modulus and shear modulus of *P* structure decrease about 90%, as the volume fraction decreases from 80% to 20%. To understand the distinguishing *E*/*S* ratio of *P* structure, stress analysis is performed as shown in Fig. 3b. It shows that stress is concentrated only on connections between basis atoms regardless of the structure type and the loading direction. When the loading is applied on *D* and *G* structures, compressive and shear stress are always generated on the connections regardless of the loading direction. In case of *P* structures, however, a compressive stress is generated only at connections in directions parallel to the loading direction as the compressive loading is applied.

However, stress is generated in all the connections when shear loading is applied. Thus, we simplify the *P*, *D* and *G* structures by beam structures shown as red beams in Fig. 3c^{27}. Beams can be described by diameter (*d*) and connection angle (θ). The connection angle is defined as the angle between beams and the *xy*-plane where the loading is applied. The connection angle of the *P* structure is 90^{o} and those of *D* and *G* structures are 45^{o}. We calculate deflections of beams under compressive and shear loadings by using Euler-Bernoulli beam theory^{28} According to the beam theory, the deflections of beams under compressive and shear loadings are inversely proportional to the second and the forth power of beam diameter (*d*), respectively. Thus, the deflection under shear loading is more significantly affected by the change of beam diameter than that under compressive loading. When…

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Read More:Controlled Unusual Stiffness of Mechanical Metamaterials – Scientific Reports