Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions
Introduction
The collision between deformable objects has been the subject of intensive investigation by many researchers using theoretical, numerical, and experimental methods (e.g., [1], [2], [3], [4], [5]). Our work is motivated primarily by the need to develop more accurate and reliable contact force–displacement (FD) models for granular flow simulations using the discrete-element method (DEM) (see [6], [9]). For elastic contact, Hertzian contact mechanics (see [7], [8]) provides an accurate nonlinear elastic model. In granular flow simulations, often much simpler (linear) models are used.
When plastic deformation is involved, the collision/contact problems become so complicated that an accurate theoretical solution is difficult to obtain. In most collisions, plastic deformation occurs, causing energy to be dissipated, and resulting in a coefficient of restitution less than unity. For elasto-plastic collisions, Walton and Braun proposed a simplified linear model based on finite element analysis (FEA) results. A more refined model was proposed in Thornton [10]. More recently, a new elasto-plastic normal FD (NFD) model based on an additive decomposition of the contact radius and a generalization of Hertzian contact mechanics to the nonlinear materials is proposed in Vu-Quoc and Zhang [11]. Even though experimental results were presented in Goldsmith [1] and in Kangur and Kleis [4], the material and geometry properties were not given in detail for use in a model (which could be either a finite element model or a force–displacement model for granular flow simulations).
In the present work, we use the nonlinear FEA code ABAQUS [12] to model the dynamic process of the collision between a deformable sphere and a rigid, frictionless planar surface. Both elastic material and elasto-plastic material are considered. The results from the elastic material are compared to Hertz contact to calibrate the FEA model. After we switch to an elasto-plastic material in the FEA model, the results are then used to compare to those obtained from the elasto-plastic FD models by Thornton [10] and by Vu-Quoc and Zhang [11]. Such a comparison can be viewed as a validation of these FE models for granular flow simulations. Note that we are considering here the case of normal frictionless impact of spheres. For oblique impacts of deformable bodies, friction plays an important role in the coefficients of restitution; we refer to Stronge [13] and Vu-Quoc et al. [5] for more details.
Section snippets
Finite element model
Fig. 1 shows a sphere colliding against a frictionless rigid planar surface, a situation equivalent to two identical spheres with the same velocity amplitude colliding against each other. In our nonlinear dynamic FEA, the size and the material properties of the sphere are chosen to be, radius , Young's modulus , Poisson's ratio ν=0.3, and density . For elasto-plastic collisions, elasto-perfectly plastic model with von Mises yield criterion is employed. The
Elastic collisions
Using the nonlinear FE code ABAQUS [12], with the FE models described in Section 2, we carry out a series of dynamic FEA for elastic collisions between an elastic sphere and a frictionless rigid surface with different incoming velocities. As mentioned in Section 1, the behavior of such collisions can be solved theoretically using Hertz theory through a quasi-static procedure. In this section, we compare our dynamic FEA results of elastic collisions with the corresponding results obtained by
Elasto-plastic collisions
In this section, we present the dynamic FEA results for elasto-plastic collisions between a sphere of elasto-perfectly plastic material and a frictionless rigid planar surface. In addition, we compare our FEA results with the results of DEM simulation using the Vu-Quoc and Zhang [11] elasto-plastic NFD model to show the correctness of the NFD model. At first, a brief introduction of the Vu-Quoc and Zhang [11] elasto-plastic NFD model is given below.
Conclusion
We presented the dynamic simulations of the collisions between a sphere and a frictionless rigid planar surface using the nonlinear finite-element code ABAQUS. Such collisions are equivalent to the collisions between two identical spheres. The results of the collisions using an elastic sphere obtained from our FEA agreed closely with the results produced by applying the Hertz theory in various aspects. Such an agreement validated the reliability of the FEA models used in our FEA. The
Acknowledgements
We thank our colleagues, Dr. Otis Walton and Mr. Lee Lesburg, for their discussions, and Mr. Tam Trinh for his help in post-processing of the FEA results. We also thank Dr. W.J. Stronge for his papers and discussion. The support of the National Science Foundation is gratefully acknowledged.
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Graduate research assistant; now with Siemens Corporate Research, Princeton, New Jersey.