One of the parameters that describe particle-particle collision is a coefficient of restitution. This can be simply defined as a ratio of the post-collisional and pre-collisional relative velocity. This chapter is devoted to this topic. As it is straightforward to measure this parameter experimentally, different practical techniques have been used by the researchers, and they are depicted here. Factors such as material properties and pre-collisional conditions are discussed, and it is shown how they influence the value of the coefficient of restitution. It is worth noting that the coefficient of restitution can also be found theoretically by exploiting the relationships previously discussed in the book, especially in Chapter 3. This is described in detail in this chapter. The chapter therefore returns to the previously considered mathematical models. Finally, the chapter concludes with two additional sections focusing on special cases: collisions of granules and nanoparticles., respectively. These particular types of particles have unique features that greatly influence the collision process and restitution coefficient.

This chapter explains the hard-sphere model of particle-particle collision. This model exploits impulse equations that directly relate the pre-collisional and post-collisional velocities of the particles. Thus, this model does not track the deformation history that was done in the prior chapters. As a result, we obtain ready analytical solutions so that the computational time is short. First, the chapter shows a standard hard-sphere model for a “mechanical” collision of two bodies. Different strategies are presented, such as the so-called two- and three-parameter hard-sphere model. Later, an extension of these models is shown that also accounts for adhesive interactions. Although, due to its simplicity, the hard-sphere model may not account for various physical phenomena between colliding particles, it may still be used in many applications. In this chapter, the reader is again provided with a computer code.

All the previous theorems and methods are adapted to percussive problems (problems with sudden changes of velocities). The main idea is to integrate them over the percussion interval. The behavior of unilateral constraints is analyzed in detail. The energy balance in percussive situations is particularly challenging. The formulation of energy dissipation through coefficients of restitution (used extensively in the scientific literature but not always in a consistent way) is discussed with a rigor unusual in many texts on classical mechanics. Multiple-point and rough impacts (impacts with friction) are introduced at the end of the chapter.