Propagation of elastic pressure waves in a beam window

As particle accelerator beam power increases, stress on beam windows and targets increases. Many simulations are carried out to model the dynamic stresses that are induced in these critical components by near instantaneous beam heating. However while it is often easy to obtain simulation results there are few analytical solutions available to check the accuracy of simulation techniques. We follow the strand of several authors over the years who have offered analytical solutions to the classic problem of radial stress waves in a beam window. Many of these significant contributions have still had niggling issues with regard to resolving peak stress and limitations on the applied initial heating condition. We formulate an analytical expression for the radial pressure waves based on a Green’s function solution of Feynman’s wave equation. A complete analysis of the problem demonstrates that a hypothesis that beam induced pressure waves are composed of a static and transient component is indeed correct. The analytical expression is shown to give stable bounded solutions with easily determined peak stress levels. Finally a comparison between analytical expression and finite element analysis of the problem yields some general guidelines that should be adhered to for achieving accurate stress wave simulations.

Figure 1

“Top hat” initial condition as employed by Sievers et al.

Figure 2

Continuous initial condition as employed by Zheng et al.

Figure 3

Gaussian initial condition as used in the method presented here.

Figure 4

Normalized total pressure at $r=0$ for the test case assuming plane stress.

Figure 5

Normalized total pressure at $r=\sigma$ for the test case assuming plane stress.

Figure 6

Normalized total pressure at $r=R$ for the test case assuming plane stress.

Figure 7

Von-Mises stress at $r=0$ for the test case assuming plane stress.

Figure 8

Pressure wave components at $t=0$ for the test case assuming plane stress.

Figure 9

Pressure wave components at $t=2\mu \mathrm{s}$ for the test case assuming plane stress.

Figure 10

Pressure wave components at $t=4\mu \mathrm{s}$ for the test case assuming plane stress.

Figure 11

Propagation of transient component of pressure for the test case assuming plane stress.

Figure 12

Comparison of plane stress model representing a foil or window and plane strain representing a long cylinder.