We describe and determine the static and dynamic (Bingham) yield stresses in an electrorheological (ER) fluid from a microstructural model. The model relates both these yield stresses to the electrostatic energy determined from the suspension capacitance matrix, which we developed previously for the dynamic simulation of an ER fluid. The static yield stress is determined from nonlinear elasticity strain-energy theory applied to an ER fluid for a variety of volume fractions and particle-to-fluid dielectric constant ratios. The static yield stress increases with the dielectric constant ratio and exhibits a maximum at 40 vol % particles for dielectric constant ratios of 4 or less. From the capacitance of the suspension we also compute the zero-frequency birefringence of the ER fluid and show that it follows a nonlinear stress-optical rule. The dynamic yield stress, as we have observed in our previous simulations, dominates the rheology of the ER fluid at large electric field strengths. At the same time the electrostatic energy of the suspension undergoes repeated slow increases, followed by rapid decreases or jumps. The connection between the dynamic yield stress and these energy jumps observed in the simulations is derived from a total energy balance of a sheared ER suspension. At low Mason number, Ma, the ratio of viscous forces to electrostatic forces, the dynamic yield stress is found to be equal to the product of the average energy jump and its frequency, and the theory is successfully tested using our dynamic simulation. Using this theory, a simple model is developed that predicts the effects of volume fraction and particle-to-fluid dielectric constant ratio on the dynamic yield stress. We find that the dynamic yield stress, like the static yield stress, increases with dielectric constant ratio and there is a maximum for a volume fraction of 40% particles as is indeed observed experimentally. With this understanding of the origin of the dynamic yield stress, we are able to predict the maximum yield stress obtainable with an ER fluid.
