Abstract
Analytical solutions detailing the propagation of longitudinal waves in slender rods subjected to a sudden increase of internal energy provide simple tools for the calculation of the temperature distribution in impacted rods as well as the resulting mechanical response. The topic is of great interest in particle accelerator technology, especially with regards to collimation systems, where beam intercepting devices can be generally approximated to one-dimensional (1D) elements potentially subjected, in accidental scenarios, to abrupt thermal energy depositions induced by the impacting particles. In this study, two finite element numerical models are presented and compared to the analytical solutions by Bertarelli, Dallocchio and Kurtyka, discussing the rapid temperature increase in slender rods due to particle beam impacts and the resulting dynamic longitudinal response. The first model is a sequentially coupled thermomechanical analysis; the second is based on a modal analysis to find the harmonic response of the system. The results indicate that phenomena neglected in analytical solutions, primarily dispersion of the longitudinal wave due to interactions with the free external surface of the rod, can be included in numerical models and can be observed in simulation results. The study further shows how numerical methods can be utilized to predict the frequencies and amplitudes of high-frequency disturbances in the longitudinal wave signal, and how these effects can be mitigated in preparation for experimental scenarios by fine-tuning the geometry of the rod and varying the duration of the pulse. This is especially useful with regards to experiments conducted in the HiRadMat facility at CERN, such as the recently conducted HRMT36 experiment, where high-frequency components can distort the signal to be observed.
Highlights
Analytical solutions detailing the propagation of longitudinal waves in slender rods subjected to a sudden increase of internal energy provide simple tools for the calculation of the temperature distribution in impacted rods as well as the resulting mechanical response
The topic is of great interest in particle accelerator technology, especially with regards to collimation systems, where beam intercepting devices can be generally approximated to one-dimensional (1D) elements potentially subjected, in accidental scenarios, to abrupt thermal energy depositions induced by the impacting particles
The dynamic response of slender rods subjected to a very fast internal temperature increase is of particular interest for high energy particle physics applications, as many beam intercepting devices are in the form of long rods of material
Summary
Subatomic particles interacting with matter transfer some of the energy to the material they interact with in the form of heat, leading to a temperature increase in the impacted material. A reference stress can be defined as follows: σref 1⁄4 EαTf: ð7Þ This value corresponds to the compressive axial stress that would be induced by a uniform temperature increase Tf, in a rod with its ends fixed. B2 a ð9Þ where B is the relevant dimension (the radius in case of cylinders) and a is the thermal diffusivity, given by hypothesis that the frequency of the system is small compared to a reference value This follows the study made by Graff [6] on the breakdown of the classical wave theory with regards to dispersive effects, showing that up to a limiting working frequency radial inertia can be neglected and the classical wave theory for longitudinal wave propagation in cylindrical rods is relevant. Longitudinal wave propagation in cylindrical elastic bar, showing effects at free boundaries
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