Computational Physics Group

Karel Matous










Shock Analysis and Optimization of Two-Layered
Cellular Materials Subject to Pulse Loading

J.C. Goetz and K. Matous

Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN, 46556, USA.


    We present the method of characteristics with mass, momentum, and energy conservation to solve the nonlinear wave equation with shock formation in a two layer one-dimensional rod made of cellular material. We show that the rigid-perfectly-plastic-locking (RPPL) model cannot predict shock formation at a material interface, so we propose an elastic-plastic-densifying (EPD) model to describe the stress-strain behavior of the cellular materials. The conditions for shock formation at a material interface are provided. We conduct a two-layer analysis to gain insights into the behavior of two layer cellular systems and to determine which material properties are most important for design. Finally, we optimize the significant parameters to reduce the length of one and two layered cellular systems with impulse and mass constraints subject to pulse loading. The results reinforce the concept of sandwich structures and show that two layer systems can achieve a 30% reduction in length over single layer ones.


    The formation of shocks in two-layer cellular systems have been analyzed using the method of characteristics with thermodynamic shock theory. The EPD material model is proposed to capture the formation of shocks at the interface between material layers. We show that the RPPL model does not allow shock formation in two layer systems. The numerical implementation was verified against existing work. The two-layer analysis demonstrates that impact layer is responsible for the majority of the energy dissipation when shocks form in both layers. The search for the optimum revealed that the higher the density of impact layer, the shorter the total system length. However, there is a cost in mass associated with the reduction in total system length. The results further reinforce the established concept of sandwich materials, where a soft core is placed between stiffer outer layers. Another finding in the optimization study is that, in all cases, the time for the material to dissipate the applied energy (minus remaining elastic energies) is the same at approximately t = 0.42 ms, meaning that a two layer configuration does not appear to improve dissipation rates.
    The method as implemented is not restricted to two layers and could be executed for more interfaces. The primary drawback to this method is that it solves every wave reflection, and the number of wave reflections increases substantially for each material interface, resulting in large computational cost. Furthermore, the method as implemented only handles piece-wise constant loading inputs and additional work is required to handle problems that involve continuously varying loads.

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2013 Notre Dame and Dr. Karel Matous