Propagation of elastic pressure waves in a beam window

T R Davenne,P Loveridge

doi:10.1103/physrevaccelbeams.19.093501

Abstract

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.

Highlights

The interaction of a particle beam with a beam window or target causes heating of the material in a time much shorter than the time required for the material to physically expand
This results in higher stress levels than would occur if the material was heated by the same amount slowly and what is generally referred to as an inertial response manifested as the propagation of pressure waves through the material
In order to highlight these rules we present an analytical expression for the one dimensional propagation of radial pressure waves in a disc and compare this with an implicit Finite element analysis (FEA) simulation

Summary

INTRODUCTION

The interaction of a particle beam with a beam window or target causes heating of the material in a time much shorter than the time required for the material to physically expand This results in higher stress levels than would occur if the material was heated by the same amount slowly and what is generally referred to as an inertial response manifested as the propagation of pressure waves through the material. The forward marching solvers employed by these explicit codes allow them to simulate shock waves where there are appreciable changes in density of the material This capability is not required for the simulation of beam windows and targets which are designed to operate in the elastic regime. A test case is used to compare the final analytical expression with results from the finite element code ANSYS with a view to providing guidance for setting up FEA models of beam induced pressure waves

FORMULATION OF THE ANALYTICAL EXPRESSION

J20ðλnÞ

COMPARISON WITH FINITE ELEMENT ANALYSIS

CONCLUSIONS