A simplified model and its asymptotic solution for the crashworthiness design of graded cellular material

Abstract

Cellular materials have strong energy absorption capacity and have been extensively used in crashworthiness design of structures. The mechanical responses of cellular materials are related to their relative density. Introducing a density gradient to cellular materials may further improve their crashworthiness. A backward design method based on a nonlinear plastic shock model was developed to meet the crashworthiness requirement of protecting a mass when impinging a graded cellular rod. For the case of constant impact force, an asymptotic solution to the density distribution of graded cellular rod is obtained in this paper. The density distributions between the explicit asymptotic solution and the exact solution numerically obtained with a fourth-order Runge-Kutta scheme have been compared. The first-order approximate solution is a linear approximation, and the second-order approximate prediction is basically consistent with the numerical solution. For the impact force history, the first-order approximate prediction has a large deviation, and the second-order approximate prediction has a slight deviation. For practical application the second-order approximate solution seems to be accurate enough. Cell-based finite element models are employed to verify the asymptotic solution. The finite element models have been constructed and computed via ABAQUS/Explicit code. Samples of graded cellular rods designed from the asymptotic solution are tested numerically. The predictions using the second-order approximate solution of density distribution is very close to the finite element results. The design using the first-order approximate solution is not accurate enough, but in fact it is not so bad. Compared the uniform density distribution, the second-order approximate prediction is good for the crashworthiness optimization. The design of density distribution for a specific impact scenario is found to be inappropriate for much high velocity impact. In summary, a case study shows that the second-order approximate solution is very close to the exact solution and it is good enough for the design. The backward design method and the asymptotic solution of density distribution are helpful to guide the crashworthiness design of graded cellular materials.