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.
Abstract
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.
Conclusions
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.