Stretching theory of Hookean metashells

Abstract

Despite being governed by the familiar laws of Hookean mechanics, elastic shells patterned with an internal structure (i.e. metashells) exhibit a wealth of unusual mechanical properties with no counterparts in unstructured materials. Here I show that much of this behavior can be captured by a real-valued analog of the inhomogeneous Schrödinger equation, with the lateral pressure experienced by the internal structure in the role of the wave function. In the fine structure limit $-$ i.e. when the length scale associated with the internal structure is much smaller than the local radius of curvature $-$ this approach reveals the existence of localized states, in which elastic deformations are prevented to diffuse away from their origin, thereby allowing the internal structure to smoothly adapt to the intrinsic geometry of the metashell. Leveraging on an analogy with scattering states in nearly free electrons, I further show that periodic metashells, obtained from the repetition of the same structural unit periodically in space, support elastic Bloch waves, corresponding to stationary periodic configurations of the internal structure and characterized by a geometry-dependent band structure. When applied to crystalline monolayers, this approach provides a generalization of the elastic theory of interacting topological defect to compressible systems.

Figures (3)

• Figure 1: Examples of metashells obtained by embedding a deployable kirigami lattice on three surfaces of revolution: i.e. a torus (blue), a spherical barrel (green) and a pseudospherical barrel (green). The latter two surfaces feature constant positive and negative Gaussian curvature respectively, while Gaussian curvature of the torus varies from positive (outside) to negative (inside).
• Figure 2: Stretching of kirigami methashells. (a) On the left, an illustration of the fundamental deformation modes of a unit cell: i.e. deployment and inflation. On the right, an example of how the underlying Gaussian curvature of a surface causes the geodesic to converge (top, $K>0$) or diverge (bottom, $K<0$). (b,c) Spatial configuration of the dimensionless later pressure $\Pi$ obtained from numerical simulations of a spring model of kirigami metashell, with spherical/pseudospherical and (b) toroidal geometry (see Sec. S5 of Ref. SI for details). The red and black lines in in panel (c) mark the positions of the external equator and the circles, located at $s= \pm L/4$, where the Gaussian curvature changes in sign. (d-f) Top and side views of the simulated metashells.
• Figure 3: Bloch waves in periodic metatubes. (a) Example of periodic tubular surface obtained upon connecting spherical and pseudospherical barrels so to guarantee the continuity of the tangent plane, while the Gaussian curvature is piecewise constant function of the geodesic latitude switching in between $-1/R_{-}^{2}$ and $1/R_{+}^{2}$. On the right, a portion of a Bloch wave solution of Eq. \ref{['eq:shape']} for $\beta=1$, $R_{-}=a$, $R_{+}/R_{-}=0.52414$ and $b/a=0.18708$ and three different $K_{0}$ values. (b) Band structure associated with the microstructure of Bloch waves for the same parameter values of panel (a). The horizontal dotted lines mark the $K_{0}$ values of the three solutions.