  Soft composite solids are made of inclusions dispersed within soft matrices.
They are ubiquitous in nature and form the basis of many biological tissues. In
the field of materials science, synthetic soft composites are promising
candidates for building various engineering devices due to their highly
programmable features. However, when the volume fraction of the inclusions
increases, predicting the mechanical properties of these materials poses a
significant challenge for the classical theories of composite mechanics. The
difficulty arises from the inherently disordered, multi-scale interactions
between the inclusions and the matrix. To address this challenge, we
systematically investigated the mechanics of densely filled soft elastomers
containing stiff microspheres. We experimentally demonstrated how the
strain-stiffening response of the soft composites is governed by the critical
scalings in the vicinity of a shear-jamming transition of the included
particles. The proposed criticality framework quantitatively connects the
overall mechanics of a soft composite with the elasticity of the matrix and the
particles, and captures the diverse mechanical responses observed across a wide
range of material parameters. The findings uncover a novel design paradigm of
composite mechanics that relies on engineering the jamming properties of the
embedded inclusions.
