# Dynamic enlargement of a hole in a sheet: crater formation and propagation of cylindrical shock waves

## TL;DR

This paper provides a complete theoretical solution for the dynamic expansion of a cylindrical hole in a strain hardening elastoplastic sheet, revealing shock wave formation at high velocities and implications for protective material design.

## Contribution

It introduces the first comprehensive theoretical model of dynamic crater formation in plane-stress conditions, including shock wave analysis and constitutive sensitivity.

## Key findings

- Shock waves occur at lower expansion velocities in plane-stress deformation.
- Material compressibility has negligible effect on shock wave formation.
- The model reduces to classical solutions in the quasistatic limit.

## Abstract

Predicting the shape of a crater formed by high velocity impact is of interest in several fields. It can aid in design of more efficient protective structures, in forensic analysis of bullet holes, and in understanding the effects of meteorite impact in both space systems and in extreme geological events. In this paper we present, for the first time, a complete theoretical solution of the dynamic plane-stress problem. We consider the steady-state expansion of a cylindrical hole in a strain hardening elastoplastic sheet and find that a self-similar field emerges if the `specific cavitation energy' is constant. It is shown that at the quasistatic limit this solution reduces to available classical solutions, while at high expansion velocities shock waves can appear. Investigation of the constitutive sensitivities of the expansion field is conducted and compared with available results for the spherical field which is commonly applied to predict resistance to high velocity penetration. It is shown that shock waves appear at significantly lower expansion velocities, in the plane-stress deformation pattern, for which material compressibility is found to have a negligible effect. This insensitivity can be taken advantage of in the future for design of light weight protective layers by incorporating porosity.