Metallic materials used in engineering applications frequently contain some volume fraction of particles. These may be impurity particles or particles that are deliberately added to influence the microstructure behavior.
As grain boundaries migrate through the microstructure, driven by boundary curvature and/or stored energy gradients, any particles present will impede - or even prevent - the movement by exerting drag forces on the passing boundaries. Classically this effect is termed Zener drag or Zener pinning. This pinning pressure is usually written as
where the negative sign indicates a retarding pressure and where is the boundary energy while and are the volume fraction of particles and the average particle size, respectively. and are parameters.
In the original model by Zener (published by C.S. Smith in 1948), the formulation is based on the assumption of rigid grain boundaries between the particles, providing . Later studies have tried to incorporate the tendency of the grain boundary to bow out between particles, arriving at other values of the parameters and .
A mesoscale RVE model of dynamic discontinuous recrystallization, considering particle drag, is established in Hallberg et al. (2014). The model is based on a 3D cellular automaton formulation. See illustration below.