Shock Waves in Solid Targets

1. Shock Waves in Solid Targets Preliminary Calculations Chris Densham

2. Codes used for study of shock waves • Specialist codes eg used by Fluid Gravity Engineering Limited – Arbitrary Lagrangian-Eulerian (ALE) codes (developed for military) • Developed for dynamic e.g. impact problems • ALE not relevant? – Useful for large deformations where mesh would become highly distorted • Expensive and specialised Chris Densham

3. Codes used for study of shock waves • Specialist codes eg used by Fluid Gravity Engineering Limited – Arbitrary Lagrangian-Eulerian (ALE) codes (developed for military) • Developed for dynamic e.g. impact problems • ALE not relevant? – Useful for large deformations where mesh would become highly distorted • Expensive and specialised • LS-Dyna • Uses Explicit Time Integration (ALE method is included) • suitable for dynamic e.g. Impact problems i.e. ΣF=ma • Should be similar to Fluid Gravity code (older but material models the same?) Chris Densham

4. Codes used for study of shock waves • Specialist codes eg used by Fluid Gravity Engineering Limited – Arbitrary Lagrangian-Eulerian (ALE) codes (developed for military) • Developed for dynamic e.g. impact problems • ALE not relevant? – Useful for large deformations where mesh would become highly distorted • Expensive and specialised • LS-Dyna • Uses Explicit Time Integration (ALE method is included) • suitable for dynamic e.g. Impact problems i.e. ΣF=ma • Should be similar to Fluid Gravity code (older but material models the same?) • ANSYS • Uses Implicit Time Integration • Suitable for ‘Quasi static’ problems ie ΣF≈0 Chris Densham

5. Implicit vs Explicit Time Integration • Explicit Time Integration (used by LS Dyna) • Central Difference method used • Accelerations (and stresses) evaluated at time t • Accelerations -> velocities -> displacements • Small time steps required to maintain stability • Can solve non-linear problems for non-linear materials • Best for dynamic problems (ΣF=ma) Chris Densham

6. Implicit vs Explicit Time Integration • Implicit Time Integration (used by ANSYS) - • Finite Element method used • Average acceleration calculated • Displacements evaluated at time t+Δt • Always stable – but small time steps needed to capture transient response • Non-linear materialscan be used to solve static problems • Can solve non-linear (transient) problems… • …but only for linear material properties • Best for static or ‘quasi’ static problems (ΣF≈0) Chris Densham

7. Study by Alec Milne Fluid Gravity Engineering Limited • “Cylindrical bar 1cm in radius is heated instantaneously from 300K to 2300K and left to expand” Chris Densham

8. Study by Alec Milne, Fluid Gravity Engineering Limited The y axis is radius (metres) Chris Densham

9. Study by Alec Milne Fluid Gravity Engineering Limited • Alec Milne: • “We find that these models predict there is the potential for a problem […]. These results use 4 different material models. All of these show that the material expands and then oscillates about an equilibrium position. The oscillations damp down but the new equilibrium radius is 1.015cm. i.e. an irreversible expansion of 150 microns has taken place. The damping differs from model to model. The key point is all predict damage.” Chris Densham

10. Study by Alec Milne Fluid Gravity Engineering Limited • Alec Milne: • “We find that these models predict there is the potential for a problem […]. These results use 4 different material models. All of these show that the material expands and then oscillates about an equilibrium position. The oscillations damp down but the new equilibrium radius is 1.015cm. i.e. an irreversible expansion of 150 microns has taken place. The damping differs from model to model. The key point is all predict damage.” • NB: 1. Thermal expansion αrΔT = 65 microns • 2. The calculation is for ΔT = 1000 K, whereas • for a Nufact target ΔT ≈ 100 K Chris Densham

11. Can ANSYS be used to study proton beam induced shockwaves? • Equation of state giving shockwave velocity: For tantalum c0 = 3414 m/s Chris Densham

12. Can ANSYS be used to study proton beam induced shockwaves? • Equation of state giving shockwave velocity: For tantalum c0 = 3414 m/s Cf: ANSYS implicit wave propagation velocity : Chris Densham

13. ANSYS benchmark study: same conditions as Alec Milne/FGES study i.e.ΔT = 1000 K The y axis is radial deflection (metres) Chris Densham

14. Comparison between Alec Milne/FGES and ANSYS results Chris Densham

15. ANSYS benchmark study: same conditions as Alec Milne/FGES study - EXCEPT ΔT = 100 K (not 1000 K) Surface deflections in 1 cm radius Ta rod over 20 μs after ‘instantaneous’ uniform temperature jump of 100 K Chris Densham

16. ANSYS benchmark study: same conditions as Alec Milne/FGES study - EXCEPT ΔT = 100 K (not 1000 K) Elastic stress waves in 1 cm radius Ta rod over 20 μs after ‘instantaneous’ (1ns) pulse Stress (Pa) at : centre (purple) and outer radius (blue) Surface deflections in 1 cm radius Ta rod over 20 μs after ‘instantaneous’ uniform temperature jump of 100 K Chris Densham

17. ANSYS benchmark study: same conditions as Alec Milne/FGES study - EXCEPT ΔT = 100 K (not 1000 K) Elastic stress waves in 1 cm radius Ta rod over 20 μs after ‘instantaneous’ (1ns) pulse Stress (Pa) at : centre (purple) and outer radius (blue) Surface deflections in 1 cm radius Ta rod over 20 μs after ‘instantaneous’ uniform temperature jump of 100 K Cf static case: = 400 x 106 Pa Chris Densham

18. Elastic shock waves in a candidate solid Ta neutrino factory target • 10 mm diameter tantalum cylinder • 10 mm diameter proton beam (parabolic distribution for simplicity) • 300 J/cc/pulse peak power (Typ. for 4 MW proton beam depositing 1 MW in target) • Pulse length = 1 ns Chris Densham

19. Elastic shock waves in a candidate solid Ta neutrino factory target Temperature jump after 1 ns pulse (Initial temperature = 2000K ) Chris Densham

20. Elastic shock waves in a candidate solid Ta neutrino factory target Elastic stress waves in 1 cm diameter Ta cylinder over 10 μs after ‘instantaneous’ (1ns) pulse Stress (Pa) at : centre (purple) and outer radius (blue) Chris Densham

21. Material model data • At high temperatures material data is scarce… • Hence, need for experiments to determine material model data e.g. • Standard flyer-plate surface shock wave experiment (difficult at high temperatures and not representative of proton beam loading conditions) • Scanning electron beam (can achieve stress and thermal cycling ie fatigue but no ‘shock’ wave generated) • Current pulse through wire (JRJB talk) • Experiment at ISOLDE (Is it representative? Can we extract useful data?) Chris Densham

22. Chris Densham

23. Elastic shock wave studies for draft ISOLDE proposal • 3 mm diameter Ta cylinder • Beam diameter = 1 mm (parabolic distribution for simplicity) • Peak power deposited = 300 J/cc • Pulse length = 4 bunches of 250 ns in 2.4 μs Chris Densham

24. Elastic shock wave studies for draft ISOLDE proposal Temperature jump after 2.4 μs pulse (Initial temperature = 2000K ) Chris Densham

25. Elastic shock wave studies for draft ISOLDE proposal Temperature profile at centre of cylinder over 4 x 250 ns bunches Chris Densham

26. Elastic shock wave studies for draft ISOLDE proposal Temperature profile at centre of cylinder over 4 x 250 ns bunches Radial displacements of target cylinder surface during and after pulse Chris Densham

27. Elastic shock wave studies for draft ISOLDE proposal Temperature profile at centre of cylinder over 4 x 250 ns bunches Elastic stress waves target rod over 5 μs during and after pulse Stress (Pa) at : centre (blue) outer radius (purple) beam outer radius (red) Chris Densham

28. Comparison between Nufact target and ISOLDE test Peak power density = 300 J/cc in both cases Temperature jump after 2.4 μs pulse (Initial temperature = 2000K ) Chris Densham

29. Effect of pulse length on shockwave magnitude Chris Densham