Title: Squish Jamming

Abstract: 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.