Dynamic crushing of cellular materials: Continuum-based shock wave models for the transitional and shock modes

Z Zhijun Zheng, Y Liu, J Yu, Stephen R Reid

Research output: Contribution to journalArticlepeer-review

117 Citations (Scopus)


As shown in the extensive studies of the dynamic responses of cellular materials, when the impact velocity is high, ‘shock’ waves can be generated. Because of the nature of the cellular structure, behind the ‘shock (or compaction) front’, there is a region of thickness approximately one single-cell-layer, across which the deformation can vary enormously, with strains of the order of ∼0.8, say. This is due to the extensive and progressive crushing of the cells. The compressed part of the cellular material is crushed and densified as the material crosses the front. Depending on the details of the cellular geometry, this locally large deformation can be very intricate to model, however, a first order ‘shock’ model can be defined, which permits a useful understanding of the phenomenology of the dynamic deformation of cellular materials, particularly metal foams.
However, when the impact velocity is not very high, there exists a different type of front behind which the strain, though plastic, does not reach the densification strain. Based on one-dimensional continuum-based stress wave theory with a ‘rigid unloading’ assumption, in this paper a theoretical framework is established to explore the corresponding inherent mechanisms as a simple extension of the original ‘shock’ theory.
Two models, namely the Shock-Mode model and the Transitional-Mode model, are introduced. The distributions of stress, strain and velocity in the foam rod are derived. The theoretical results show that for a Shock Mode, behind the front the initial strain remains constant and the initial stress varies proportionally with the square of the impact velocity, but for a Transition Mode, the initial strain and stress behind the front reduce linearly with reducing impact velocity. The critical impact velocities for modes transition are predicted. Two dimensionless parameters, namely the shock-enhancement parameter and the stress-hardening parameter, are defined and the features of the theoretical predictions are presented. Compared to the experimental results, the responses at the ends of foam rod are well predicted by the present models and also by the R-P-P-L model. However, deformation mechanisms uncovered by the present modes and the R-P-P-L model are very different when the impact velocity is not very high.
The present simple, wave-based models extend the understanding of metallic foams to loading over a wider range of impact velocities than the previous models. In particular, the sub-shock-like behaviour, which has not yet been dealt within the literature, can be better understood through the new Transitional-Mode model.
Original languageEnglish
Pages (from-to)66-79
Number of pages14
JournalInternational Journal of Impact Engineering
Early online date3 Oct 2011
Publication statusPublished - Apr 2012

Bibliographical note

This work is supported by the National Natural Science Foundation of China (Projects Nos. 90916026, 11002140 and 90205003), the China Postdoctoral Science Foundation (Project No.
20100470860) and the Chinese Academy of Sciences (Grant No. KJCX2-EW-L03). The authors wish to express their appreciation to Dr. P.J. Tan and Dr. J.J. Harrigan for useful and helpful communications, and to Professor T.X. Yu for facilitating their collaboration on this work.


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