Document Type


Date of Degree

Fall 2016

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

David E. Stewart

First Committee Member

Keith Stroyan

Second Committee Member

Colleen Mitchell

Third Committee Member

Gerhard Strohmer

Fourth Committee Member

Bruce Ayati


We simulate the behavior of a steel ball bearing as it impacts a rigid foundation by solving a discretized version of the dynamic equations of linearized elasticity for a homogeneous, isotropic material. Space is discretized using the finite element method and time is discretized using the implicit trapezoidal method. Impact with a fixed foundation is incorporated into the model using a complementarity condition. This ensures that we have normal forces acting on the bearing only when and where the bearing is in contact with the foundation. After discretization in space, this condition becomes a linear complementarity problem which is solved using an iterative method for solving LCPs that is similar to the Gauss-Seidel method for solving linear systems. The LCP is solved at each time step to determine the normal forces due to contact. By assuming cylindrical symmetry, we are able to simulate the impact of a three-dimensional ball using only two spatial coordinates and two-dimensional finite elements. This decreases the computational cost of a highly refined three-dimensional simulation dramatically. Using this model, we investigate the deformations that occur during and after contact. We hypothesized that dropping a steel ball from even a small height causes plastic deformation. We tested this hypothesis using our model by computing the state of stress inside the ball at various times during the simulation. By comparing the computed maximum shear stress to the yield strength of the material, we can determine if the threshold for plastic deformation is reached. We found that with an impact speed of 2 m/s the stresses induced in the ball are large enough to cause plastic deformation. Because plastic deformation requires energy and is irreversible, it is an important consideration when investigating how high the ball will bounce after contact. To quantify the energy loss due to plastic deformation, we propose a theoretical model capable of describing plastic deformation.

Public Abstract

Can you drop the same ball bearing twice? When a ball bearing impacts a rigid surface, huge forces act on the surface of the ball. These forces prevent the ball from passing through the surface and ultimately cause the ball to change direction. Such collisions are surprisingly violent, even at low speeds. They result in the production of light, heat, sound, elastic vibrations, and possibly plastic deformation. Plastic deformation refers to the permanent change of the shape of the ball in response to forces. The reality of these phenomena explains the fact that a dropped ball can never bounce to the height from which it was dropped. To further investigate this fact, we generated high-resolution simulations of steel ball bearing impacts. Our primary goal was to determine the relative importance of plastic deformation in low speed collisions. These simulations show that a drop from a modest height is enough to permanently deform a steel ball bearing. We present visualizations of the forces acting on and within the ball during impact and compare the results of our simulations with classical investigations of contact mechanics. Our model largely agrees with classical theory and provides further evidence of the usefulness of the predictions made by Hertzian contact theory. Additionally, we provide an estimate of the amount of energy lost to plastic deformation. This estimate and the small amount of vibrational energy observed in our simulations points to plastic deformation as the primary energy sink in low speed steel ball collisions. Further investigations in which complex interactions such as friction and plastic deformation are added to the model are planned for the future.


elasticity, Impact, LCP


viii, 62 pages


Includes bibliographical references (pages 61-62).


Copyright © 2016 Benjamin M. Dill