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We derive from particle-level dynamics a constitutive model describing therheology of two-dimensional dense soft suspensions below the jammingtransition, in a regime where hydrodynamic interactions between particles arescreened. Based on a statistical description of particle dynamics, we obtainthrough a set of physically plausible approximations a non-linear tensorialevolution equation for the deviatoric part of the stress tensor, involving thestrain-rate and vorticity tensors. This tensorial evolution equation involvessingular terms usually not taken into account in phenomenological constitutivemodels, which most often assume a regular expansion in terms of the stresstensor. All coefficients appearing in the equation have known expressions interms of the microscopic parameters of the model. The predictions of thismicroscopically grounded constitutive model have several qualitative featuresthat are specific to the rheology of soft suspensions measured in experimentsor simulations. The model shows a typical behavior of polymeric visco-elasticmaterials, such as normal stress differences quadratic in the shear rate$\dot\gamma$, as well as typical behaviors of suspensions of stiff particles,such as a particle pressure linear in $\dot\gamma$ and a zero-shear viscositydiverging at the jamming transition. The model also predicts a sharper shearthinning than other visco-elastic models at small shear rates, in qualitativeagreement with experimental observations. Furthermore the shear thinningfollows a critical scaling close to the jamming transition.

Microscopically grounded constitutive model for dense suspensions of soft particles below jamming