  Soft, amorphous solids such as tissues, foams, and emulsions are composed of
deformable particles. However, the effect of single-particle deformability on
the collective behavior of soft solids is still poorly understood. We perform
numerical simulations of two-dimensional jammed packings of explicitly
deformable particles to study the mechanical response of model soft solids. We
find that jammed packings of deformable particles with excess shape degrees of
freedom possess low-frequency quartic vibrational modes that stabilize the
packings even though they possess fewer interparticle contacts than the nominal
isostatic value. Adding intra-particle constraints can rigidify the particles,
but these particles undergo a buckling transition and gain an effective shape
degree of freedom when their preferred perimeter is above a threshold value. We
find that the mechanical response of jammed packings of deformable particles
with shape degrees of freedom differs significantly from that of jammed
packings of rigid-shape particles, which emphasizes the importance of particle
deformability in modelling soft solids.
