Constitutive relations for colloidal gel

The theoretical treatment of depletion gels with central interactions often involves expanding the free energy around a stress-free reference state to derive a constitutive relation between global stress and strain. The premise upon which the previous continuum theories are based, i.e., the stress-free reference state and the affine deformation, both of which do not hold in the context of amorphous gel materials. Gels never reach a true global minimum in the potential energy landscape and contain local regions of significant compressive and tensile stress, interspersed with zero-stress regions. Hence, expansion of free energy around a stressed reference state will produce scalar terms in harmonic expansion, the effects of which are qualitatively different from the terms appearing in the expansion around an unstressed reference state. In this study, we demonstrate the limitations of traditional continuum theories and propose simple constitutive relations that better capture the mechanical response of gel materials. The robustness of the proposed relations is established through large-scale numerical simulations of depletion and frictional gels across a vast parameter space.