  A wide range of disordered materials, from biological to geological
assemblies, feature discrete elements undergoing large shape changes. How
significant geometrical variations at the microscopic scale affect the response
of the assembly, in particular rigidity transitions, is an ongoing challenge in
soft matter physics. However, the lack of a model granular-like experimental
system featuring large and versatile particle deformability impedes advances.
Here, we explore the oscillatory shear response of a sponge-like granular
assembly composed of highly compressible elastic rings. We highlight a
progressive rigidity transition, switching from a fluid-like to a solid-like
response by increasing density or decreasing shear amplitude. The rearranging
fluid state consists of crystal clusters separated by melted regions; in
contrast, the solid state remains amorphous and absorbs all imposed shear
elastically. We rationalise this transition by uncovering an effective,
attractive shear force between rings that emerges from a friction-geometry
interplay. If friction is sufficiently high compared to shear, the extent of
the contacts between rings, captured analytically by elementary geometry,
controls the rigidity transition.
