AUTHOR=Reddy A. , Karami A. , Nassar H. 

TITLE=Frustration propagation in tubular foldable mechanisms

JOURNAL=Frontiers in Physics

VOLUME=11

YEAR=2023

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2023.1296661

DOI=10.3389/fphy.2023.1296661

ISSN=2296-424X

ABSTRACT=<p>Shell mechanisms are patterned surface-like structures with compliant deformation modes that allow them to change shape drastically. Examples include many origami and kirigami tessellations as well as other periodic truss mechanisms. The deployment paths of a shell mechanism are greatly constrained by the inextensibility of the constitutive material locally and by the compatibility requirements of surface geometry globally. With notable exceptions (e.g., Miura-ori), the deployment of a shell mechanism often couples in-plane stretching and out-of-plane bending. Here, we investigate the repercussions of this kinematic coupling in the presence of geometric confinement, specifically in tubular states. We demonstrate that the confinement in the hoop direction leads to a frustration that propagates axially, as if by buckling. We fully characterize this phenomenon in terms of amplitude, wavelength, and mode shape in the asymptotic regime, where the size of the unit cell of the mechanism <italic>r</italic> is small compared to the typical radius of curvature <italic>ρ</italic>. In particular, we conclude that the amplitude and wavelength of the frustration are of order <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow><mml:mi>r</mml:mi><mml:mo>/</mml:mo><mml:mi>ρ</mml:mi></mml:mrow></mml:msqrt></mml:math></inline-formula> and that the mode shape is an elastica solution. Derivations are carried out for a particular pyramidal truss mechanism. The findings are supported by numerical solutions of the exact kinematics.</p>