Three-dimensional mechanical metamaterials with a twist

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Science  24 Nov 2017:
Vol. 358, Issue 6366, pp. 1072-1074
DOI: 10.1126/science.aao4640
  • Fig. 1 Twist degrees of freedom in mechanics.

    (A) Pushing on an elastic material bar (red arrow) can make it expand or contract isotropically or anisotropically in the orthogonal directions. (B) A twist, however, is forbidden in ordinary linear (Cauchy) continuum mechanics. (C) Unit cell of a metamaterial crystal enabling the twist degree of freedom. The lattice constant a, the angle δ, the radii r1 and r2, and the widths b and d are indicated. (D) Calculated deformed cell and displacement under uniaxial loading. The arrows aid the discussion of the mechanism: 1. The arms connecting the corners with the rings move downward. 2. This motion leads to a rotation of the rings. 3. This rotation exerts forces onto the corners in the plane normal to the pushing axis, resulting in an overall twist of the unit cell around this axis (also see fig. S1).

  • Fig. 2 Gallery of electron micrographs.

    (A to E) Polymer samples following the blueprint shown in Fig. 1C, fabricated using 3D laser microprinting. Arranging a left-handed metamaterial bar on top of a right-handed one enables twists without the need for sliding boundary conditions. (G to K) Changing the number of unit cells within the bars, while fixing all aspect ratios and outer dimensions, is crucial to investigate the breakdown of scalability associated with mechanical chirality. (F) and (L) are achiral controls. The measured azimuthal displacement vectors (blue) added in (C) indicate a twist upon pushing on the metamaterial bar. The axial displacement vectors are shown as red arrows (movie S1). All arrows are stretched by a factor of 5.

  • Fig. 3 Measured and calculated results.

    We vary the total number of unit cells (N × N × 2N) × 2 with N = L/a = 1,2,3,4,5 while fixing the outer dimensions of the samples (compare Fig. 2). (A) Twist angle per axial strain versus N. (B) Effective Young’s (E) modulus of the metamaterial bar versus N. Red (blue) symbols correspond to chiral (achiral) samples. Circles (0.5% strain), squares (1.0% strain), and triangles (1.5% strain) are measured. The statistical error bars are on the scale of the size of the symbols. The crosses are calculated in the linear regime. The solid red curves result from micropolar continuum mechanics (28) (Fig. 4B); the solid blue straight lines are the expectation from Cauchy continuum mechanics.

  • Fig. 4 Finite-element calculations.

    Structure and modulus of the displacement vector field on a false-color scale are overlaid. (A) Microstructure calculation for N = 3 (compare Fig. 2C). (B) Calculation following chiral micropolar continuum mechanics for L = 3a with a = 500 μm (28). In both cases, the axial strain is 1%. The black lines indicate the sample boundaries for zero strain. For better visibility, the deformations have been multiplied by a factor of 10. The false-color scale shows the true displacements. All other parameters are as in Fig. 3 (28).

Supplementary Materials

  • Three-dimensional mechanical metamaterials with a twist

    Tobias Frenzel, Muamer Kadic, Martin Wegener

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Supplementary Text
    • Figs. S1 to S7
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    Images, Video, and Other Media

    Movie S1
    The following link leads to an optical microscopy movie of the experiment for N = 3. The left part of the movie exhibits a bottom view of the sample onto the plate in between the left- and right-handed part of the sample (compare Fig. 2C). The right part is a side view onto the same sample. Upon pushing onto the sample, one can see a rotation around the pushing axis on the left and a compression along the pushing axis on the right-hand side. The resulting rotation angle divided by the axial strain is depicted in Fig. 3A for different samples. In contrast to Fig. 2C, all arrows are shown on the same scale as the images in the background.

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