Richtmyer–Meshkov instability for elastic–plastic solids in converging geometries

• Published in: Journal of the mechanics and physics of solids 2015-03, Vol.76, p.291-324
• Main Authors: López Ortega, A., Lombardini, M., Barton, P.T., Pullin, D.I., Meiron, D.I.
• Format: Article
• Language: English
• Subjects: Deformation; Density; Determinants; Eulerian algorithm; Failure; Hyper-elastic constitutive laws; Instability; Inversions; Mathematical analysis; Mathematical models; Richmyer–Meshkov instability; Shocks in solids; Tensors
• ISSN: 0022-5096
• DOI: 10.1016/j.jmps.2014.12.002

We present a detailed study of the interface instability that develops at the boundary between a shell of elastic–plastic material and a cylindrical core of confined gas during the inbound implosive motion generated by a shock-wave. The main instability in this configuration is the so-called Richtmyer–Meshkov instability that arises when the shock wave crosses the material interface. Secondary instabilities, such as Rayleigh–Taylor, due to the acceleration of the interface, and Kelvin–Helmholtz, due to slip between solid and fluid, arise as the motion progresses. The reflection of the shock wave at the axis and its second interaction with the material interface as the shock moves outbound, commonly known as re-shock, results in a second Richtmyer–Meshkov instability that potentially increases the growth rate of interface perturbations, resulting in the formation of a mixing zone typical of fluid–fluid configurations and the loss of the initial perturbation length scales. The study of this problem is of interest for achieving stable inertial confinement fusion reactions but its complexity and the material conditions produced by the implosion close to the axis prove to be challenging for both experimental and numerical approaches. In this paper, we attempt to circumvent some of the difficulties associated with a classical numerical treatment of this problem, such as element inversion in Lagrangian methods or failure to maintain the relationship between the determinant of the deformation tensor and the density in Eulerian approaches, and to provide a description of the different events that occur during the motion of the interface. For this purpose, a multi-material numerical solver for evolving in time the equations of motion for solid and fluid media in an Eulerian formalism has been implemented in a Cartesian grid. Equations of state are derived using thermodynamically consistent hyperelastic relations between internal energy and stresses. The resolution required for capturing the state of solid and fluid materials close to the origin is achieved by making use of adaptive mesh refinement techniques. Rigid-body rotations contained in the deformation tensor have been shown to have a negative effect on the accuracy of the method in extreme compression conditions and are removed by transforming the deformation tensor into a stretch tensor at each time step. With this methodology, the evolution of the interface can be tracked up to a point at which numerical co