Normal Collision of Elastic Spheres

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

Consider a collision between two frictionless elastic spheres so that only normal force and normal velocities are involved i. e., no tangential forces or tangential velocities need to be accounted for in this case. A general case is shown in Fig. 2.15, where two spheres with different sizes, velocities, and material properties collide with each other. Only the collisional force is considered. 

From Eq. (2.75), the corresponding maximum radius of collisional contact area can be given by 

The maximum pressure in a collision can also be given based on Eq. (2.70) as 

The collisional contact duration can be obtained by rearranging Eq. (2.130) to the form 

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