Elastic-Plastic Behavior of Metal Rings During High-Speed Electromagnetic Compression

Abstract In Magnetized Target Fusion (MTF), fusion conditions are achieved by high-speed compression of magnetized plasma within a flux conserver. General Fusion is developing its MTF technology by running experiments using solid lithium cylindrical shells. The rapid compression is induced by a high-current theta-pinch of the shell. Concentric and symmetric compression of the shell must be achieved to conserve magneto-hydrodynamic stability of the plasma during compression and to reach fusion conditions. General Fusion has operated compressors for both lithium rings and cylinders that achieved various levels of symmetry before becoming asymmetric deep into compression, formulating the need to understand the dynamics of collapsing shells. The focus of this study lies on fundamental research of the elastic-plastic behavior of the cylinder during compression in vacuum. The current work aims to provide a mathematical basis for further research and advice on experimental operations, by comparing both analytical and numerical results of two established plasticity models. The problem is tackled by starting from the very basics of solid mechanics. The cylindrical shell is first approximated by a rotationally symmetric ring of uniform thickness on which a compressive dynamic radial body force is exerted. Starting with an isotropic linear elastic model, this results in a system of ordinary differential equations with time as the only independent variable. Plasticity is introduced using an isotropic linear hardening model, which is later substituted by the Johnson-Cook hardening model. After this, the model is extended to include the azimuthal angle (in polar coordinates) as independent spatial variable. The cylindrical shell is now described as a ring of varying thickness, such that it can capture asymmetry. The resulting system of equations consists of partial differential equations. For this a model with linear hardening is presented, using similar numerical methods as for the symmetric ring. The problem is solved numerically in Matlab, and for the symmetric ring problem also analytically for the linear elastic and linear hardening model. The numerical methods used in this study consist mostly of central and forward difference approximations. The use of low-order finite-difference methods allows for quick solutions and the possibility to explain, compare and validate the results. Good correspondence is found between the analytical and numerical results. A validation against the experiments done at General Fusion is not complete, but the models show similar ring compression trajectories to the experiments, highlighting the potential for future use. Both the symmetric and the asymmetric approximation of the problem give great insight into the essence of the elastic-plastic behavior of the lithium rings and cylinders, as well as the (material) parameters and driving force. The numerical results are smooth, despite the complex nonlinear nature of the problem. For the asymmetric ring problem, some improvements on the described models are proposed, as they either give questionable results or require unrealistic input parameters. In conclusion, the presented models and obtained results lay the foundation for further research into the dynamic compression of metal rings and cylindrical shells.