Figure 1

Schematic of the crossing of characteristics in a ramp wave, signaling the formation of a shock wave.Reuse & Permissions

Figure 2

Ideal scaled loading histories for Al, Cu, and polyethene.Reuse & Permissions

Figure 3

Ideal scaled loading histories for Al, Cu, and polyethene (log space).Reuse & Permissions

Figure 4

Ideal scaled loading rate as a function of instantaneous pressure, for Al, Cu, and polyethene.Reuse & Permissions

Figure 5

Ideal scaled loading rate as a function of instantaneous pressure, for Al, Cu, and polyethene. The abscissa is that of the previous graph, divided by the ordinate. This gives a convenient measure of the systematic trend of the overall pulse duration for ideal ramp loading to a given pressure, but not an accurate estimate because the ideal shape is nonlinear with monotonically increasing rate.Reuse & Permissions

Figure 6

Pressure histories from spatially resolved continuum dynamics simulation, showing shock formation in Al. The sample was driven using the ideal loading history, to $200\mathrm{GPa}$, with $l_0=100\mu \mathrm{m}$. Pressure histories are shown for Lagrangian positions at intervals of $10\mu \mathrm{m}$ from the loading surface. The shock formed simultaneously over the full pressure range at the $100\mu \mathrm{m}$ position, though with evidence of shock smearing at the ends of the pressure range. The small pressure drop after shock formation is caused by the difference in compression between isentropic and shock compression.Reuse & Permissions

Figure 7

Schematic of shock formation in pressure–particle-speed space, showing difference in particle speed that leads to partial unloading of the ramp-loaded material. For material loaded by the ramp wave, states move up the principal ramp adiabat (i.e., through the initial state of the material) to the peak ramp drive pressure. When the shock forms, a shock of the same pressure as the peak of the ramp would have a larger particle speed. The state in material loaded by a single shock lies on the principal Hugoniot. On formation of the shock, ramp-loaded and shock-loaded material must be at the same pressure and particle speed, so the ramp-loaded material reexpands down the release adiabat, and the shock pressure is slightly lower than the peak of the ramp.Reuse & Permissions

Figure 8

Schematic cross section of ramp loading experiment, showing lateral release propagating into ramp-loaded material. The edge release interacts with the continued application of the drive to produce a region of two-dimensional deformation that pinches off the one-dimensional ramp as time progresses. Solid contours in the one-dimensional region represent the ramp pressure, increasing with proximity to the drive. Thicker arrows show the direction of propagation of the waves. At any given time, the two-dimensional region has spread further laterally than axially because lateral propagation is through material of increasing sound speed.Reuse & Permissions

Figure 9

Scaled edge release at the drive surface, for ideal ramp loading. The scaled release distance is the lateral distance affected by the edge release, divided by the axial shock formation distance.Reuse & Permissions

Figure 10

Spatially resolved Lagrangian continuum dynamics simulation of edge release during ramp loading of Al. Loading was applied over a region $200\mu \mathrm{m}$ in diameter at the left side (shown by the gray line), ramping linearly to zero over $5\mu \mathrm{m}$. The loading history was chosen to give a shock formation distance of $100\mu \mathrm{m}$. The compression wave moves from left to right. The radius of the drive region was chosen to be equal to the shock formation distance, so that the scaled edge release distance is the fractional radius. This frame is at $9.8\mathrm{ns}$ after the start of loading, when the drive pressure has reached $50\mathrm{GPa}$. Upper half shows pressure contours 0.1, 0.2, 0.5, 1, 2, 5, 10, and $20\mathrm{GPa}$ increasing from right to left. Lower half shows contours of the lateral (outward) component of particle velocity: 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and $1\mathrm{km}∕\mathrm{s}$. Exact powers of 10 are thicker lines. The furthest advanced point is at the same radius as the edge of the drive region. In the drive region, edge release is approximately 55% of the way to the center, in agreement with the scaled edge release calculation. The simulations used a triangulated mesh with side lengths initially $2\mu \mathrm{m}$.Reuse & Permissions