# Dynamic crushing of cellular materials: a particle velocity-based analytical method and its application

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## Abstract

Cellular material under high-velocity impact exhibits a typical feature of layerwise collapse. A cell-based finite element model is employed herein to simulate the direct impact of a closed-cell foam, and one-dimensional velocity field distributions are obtained to characterize the crushing band propagating through a cellular material. An explicit expression for the continuous velocity distribution is derived based on the features of the velocity gradient distribution. The velocity distribution function is adopted to determine the dynamic stress–strain states of cellular materials under dynamic loading. The local stress–strain history distribution reveals that sectional cells experience a process from the precursor elastic behavior to the shock stress state, passing through the dynamic initial crushing state. A power-law relation between the dynamic initial crushing stress and the strain rate is established, which confirms the strain rate effect of cellular materials. By extracting the critical points immediately before the unloading stage in the local dynamic stress–strain history curves, the dynamic stress–strain states of cellular materials are determined. They exhibit loading rate dependence but are independent of the initial impact velocity. Furthermore, with increase of the relative density, the dynamic hardening behavior of the cellular specimen is enhanced and the crushing process event is advanced. The particle velocity-based analytical method is applied to analyze the dynamic responses of cellular materials. This method is better than continuum-based shock models, since it does not require a preassumed constitutive relation. Therefore, the particle velocity-based analytical method proposed herein may provide new ideas to carry out dynamic experimental measurements, which is especially applicable to inhomogeneous materials.

## Keywords

Cellular material Velocity distribution Dynamic behavior Strain rate effect Dynamic stress–strain state## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11802002, 11772330, and 11372308), the Fundamental Research Funds for the Central Universities (Grant WK2480000003), and the Youth Foundation of Anhui University of Technology (Grant RD17100204).

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