Elasticity of soft matter stabilized by non-spring-like forces

The mechanics of many soft matter systems are closely related to spring networks. If elements interact exclusively via pairwise, central, finite-ranged forces – as in a spring network – then the system is floppy/fluid-like below a critical coordination known as the isostatic point. However, many soft materials possess additional interactions that rigidify/jam the system below the isostatic point. These “non-spring-like forces” have various and unrelated physical origins – here I consider pre-tensioning, fiber bending in biopolymer networks, static friction in jammed granular matter, and an elastic matrix in particle-filled composites. Using these four model systems, I will show that the elasticity of soft matter stabilized by non-spring-like forces has universal features, independent of the forces’ origin. These features manifest as scaling relations in the shear modulus, which can be explained by relating them to floppy modes that appear in the limit where non-central forces are "turned off."