 The pinning mechanism is based on the fact that the amount of grain boundary area is reduced when a particle is located on a grain boundary.It is also assumed that particles are spherical and the particle-matrix interface is incoherent.When a moving grain boundary meets a particle at an angle β, the particle exerts a pinning force F on the grain boundary that is equal to F = 2πσcosβsinβ; with r the particle radius and σ the energy per unit of grain boundary area.