A microscopic Gibbs field model for the macroscopic yielding behaviour of a viscoplastic fluid

We present a Gibbs random field model for the microscopic interactions in a viscoplastic fluid. The model has only two parameters which are sufficient to describe the internal energy of the material in the absence of external stress and a third parameter for a constant externally applied stress. The energy function is derived from the Gibbs potential in terms of the external stress and internal energy. The resulting Gibbs distribution, over a configuration space of microscopic interactions, can mimic experimentally observed macroscopic behavioural phenomena that depend on the externally applied stress. A simulation algorithm that can be used to approximate samples from the Gibbs distribution is given and it is used to gain several insights about the model. Corresponding to weak interactions between the microscopic solid units, our model reveals a smooth solid-fluid transition which is fully reversible upon increasing/decreasing external stresses. If the interaction between neighbouring microscopic constituents exceeds a critical threshold the solid-fluid transition becomes abrupt and a hysteresis of the deformation states is observed even at the asymptotic limit of steady forcing. Quite remarkably, in spite of the limited number of parameters involved, the predictions of our model are in a good qualitative agreement with macro rheological experimental results on the solid-fluid transition in various yield stress materials subjected to an external stress.