Pre-stressed networks as a model for nearly floppy soft matter

Many soft matter systems undergo a transition between a fluid-like “floppy” phase and a solid-like “rigid” phase. In the floppy phase, the system can deform without energetic cost, because there are insufficient constraints on motion, e.g. contacts between particles or crosslinks between fibers. However, many floppy systems can be rigidified without increasing their mean coordination. Fiber networks provide a classic example: they are rigid with a finite bending modulus but become floppy if the modulus is turned off. Fiber networks are just one example of “nearly floppy solids” – under-coordinated networks rigidified by a perturbation that couples to floppy motions. I will discuss multiple examples of nearly floppy solids and demonstrate universal features of their mechanical and vibrational properties.