Universal elasticity in nearly floppy soft matter

In many soft matter systems, mechanics is governed by the balance between microscopic degrees of freedom and internal constraints on motion. Constraints on relative motion parallel to the bond between two particles, or between two nodes in a network, can be modeled with springs. However, other, non-central force constraints are often present as well — examples include constraints on sliding in frictional granular media and composites, constraints on bending in biopolymers and covalent glasses, and constraints on shape distortions in confluent tissues, foams, and emulsions. We consider the case where non-central force constraints are present, but carry a weaker energetic cost than their spring-like counterparts. We will show that the elasticity of these systems has universal features, independent of the origin of the non-central force constraint. These features can be explained in terms of the floppy modes that appear in the limit where non-central forces are "turned off." We derive scaling relations for the shear modulus and validate them numerically in three model systems — pre-tensioned spring networks, semi-flexible fibre networks, and composite packings of hard and soft particles.