Fig. 5: Phase diagram of the mechanical responses of soft composite solids and granular suspensions.

The G_m = 0 plane represents the suspensions consisting of particles dispersing in uncrosslinked polymers. The solid red line in the G_m = 0 plane signifies the shear-jamming transition (ϕ_J(ε)) of dense suspensions³⁷. The 3D space defined by G_m > 0 represents soft composites consisting of particles dispersing in crosslinked polymeric elastomers. The mechanical properties of dense soft composites under different strains ε are controlled by the scalings (Eq. (8)) near the critical line ϕ_J(ε). The dashed red lines G_m/G_p = ∣1−ϕ/ϕ_J(ε)∣^Δ indicate the crossover boundary from the matrix- or particle-dominated regime to the critical regime. The solid arrow (A → B → C) illustrates a representative strain-stiffening process of soft composites with a particle volume fraction ϕ_m < ϕ < ϕ₀. With the increase in the applied strain ε, the mechanical response of the composites crosses over from the matrix-dominated regime (ε = 0) to the critical regime (ε = ε₁), and finally to the particle-dominated regime (ε = ε₂).