A constitutive model for discontinuous shear thickening in epithelial tissues
The rheological properties of biological tissues, though fundamental to many physiological and pathological processes such as embryonic development, wound healing, and tumor progression, remain poorly understood. A recent study showed that the active vertex model of biological tissues exhibits discontinuous shear thickening (DST), where stress and viscosity suddenly increase at a critical shear rate. What is the mechanism of DST here? Is it another nontrivial feature of activity or an inherent property of the system? To address this, we show that the thermal vertex model also exhibits DST at a small but non-zero temperature $T$. Solid-like and liquid-like cells coexist at the stress jump, and the stress-controlled flow curves exhibit the characteristic S-shape. We then introduce a constitutive model for DST in epithelial tissues. As $p_0$ increases, the theory predicts DST, followed by continuous shear thickening (CST), and finally Newtonian behavior, consistent with simulations. DST begins at the jamming point, $p_0^m$, and the Newtonian behavior starts at $p_0^*$, where the yield stress vanishes. Both $p_0^*$ and the liquid-to-solid transition stress, $\sigma^*$, govern the DST-CST boundary. Furthermore, $p_0^*$ and $\sigma^*$ also depend on $T$. Increasing $T$ reduces $p_0^*$, narrows the shear-thickening regime, and eventually destroys DST when $p_0^* \leq p_0^m$. Thus, the primary ingredients of DST in tissue models are a finite yield stress in the unjammed regime and non-zero fluctuations, whose specific form is not important. The theory agrees well with our simulation data and also provides further testable predictions.