Instability of thin sheets under torque-free geometrically-incompatible confinement

Geometrically-incompatibe confinement (GIC) of thin sheets and shells, namely, imposing a non-zero Gaussian curvature different from the
natural ("target") one, often gives rise to morphological instabilities. GIC may be classified into ``strong'' -- where a finite difference from the natural Gaussian curvature is imposed, and ``weak'' -- whereby this difference may vanish with the body thickness. A notable example of weak GIC is the ``d-cone'' -- obtained upon pushing a disk into a ring -- where deformation is asymptotically developable, such that Gaussian curvature vanishes outside a strained core. This example underlies a common perception that under generic weak GIC sheets deform to a ``piecewise-developable'' shape.

We study a weak GIC problem, generated by a contractile inclusion, which gives rise to buckling instability. Nevertheless, in contrast to the d-cone problem, the absence of external torque allows retention of up-down symmetry. Our simulations show a far-from-threshold response strictly distinct from the wrinkle patterns characteristic of strong GIC, as well as from any developable shape. This suggests a novel class of instabilities, induced by ``ultra-weak'' GIC, where asymptotic isometry comprises a non-wrinkly yet non-developable deformation.