A finite element approach for simplified 2D nonlinear dynamic contact/impact analysis

Abstract

In this paper, a simplified numerical approach for finite element dynamic analysis of an inelastic solid structure subjected to solid object impact is presented. The approach approximates the impacting solid as the selected multiple nodes, for which mass of the impactor is distributed. The node-to-segment contact formulation with the penalty constraint technique incorporated is employed to impose contact conditions between the nodes and the surface of the receiver structure. The node-to-segment algorithm is integrated into Newton–Raphson time integration scheme and the Lagrange multiplier technique is applied to enforce the identical displacements for the selected nodes throughout the analysis process. The approach is verified using two-dimensional plane strain models considering elastic-perfectly-plastic material behavior. The results obtained using the proposed approach are in a good agreement with those simulated using a commercial finite element code, ABAQUS dynamic/implicit, in terms of displacements and stress distribution fields. The proposed approach is shown to be computationally superior to general finite element method-based contact/impact analysis without significantly sacrificing the accuracy.

Appendix

1.1 A Sensitivity analysis of penalty parameter

Many studies (for example, [2, 12, 13, 17, 24]) have been extensively conducted to find the optimum value or range for the penalty parameter (in Eq. 2) that ensures reliable and accurate analysis results. Comparison of the suggested ranges for the penalty parameter in these studies shows the penalty parameter can differ by at most the order of magnitude $10^7$ times depending upon the used materials, contact geometries, element types, etc. Unfortunately, no universal analytical expression for determining appropriate penalty parameter value exists. Therefore, this study carried out sensitivity analysis for the choice of penalty stiffness.

The penalty stiffness values were adjusted proportional to the Young’s modulus of the receiver material, as $\epsilon _N=\kappa E^{\textrm{rec}}$, where $E^{\textrm{rec}}$ is the Young’s modulus of the receiver material and $\kappa$ is the associated scale factor. A wide range of $\kappa$ was considered. To this end, $\kappa$ was set to increase 10 times for each individual run from $10^{-2}$ to $10^{+2}$. The indentation model (Sect. 3.1.2) was used for this sensitivity test. Figure 18 shows results of the computed impact force and displacement of the node in impact for different $\kappa$ values considered. As expected, both the impact force and the local nodal displacement were very sensitive to variation in the penalty parameter. Out of the five simulation runs, the run with $\kappa =1$ (i.e., $\epsilon _N=E^{\textrm{rec}}$) provided the force and displacement histories the most comparable to the ABAQUS results. It should be mentioned that a more accurate result was obtained with $\kappa =1.1$ but the difference was not significant when compared with $\kappa =1$.

Fig. 18

Sensitivity analysis results of the penalty parameter $\epsilon _N$: a Impact force and b y-direction (vertical) displacement of the receiver node in impact