Numerical simulation of the impact of a steel ball with a rigid foundation

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.