uknfwg 12 january 2005chris densham shock waves in solid targets preliminary calculations

Shock Waves in Solid Targets

Preliminary Calculations

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

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?)

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

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)

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 materials can 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)

Study by Alec Milne Fluid Gravity Engineering Limited

“Cylindrical bar 1cm in radius is heated instantaneously from 300K to 2300K and left to expand”

The y axis is radius (metres)

Study by Alec Milne, Fluid Gravity Engineering Limited

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.”

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 microns2. The calculation is for ΔT = 1000 K, whereas

for a Nufact target ΔT ≈ 100 K

Can ANSYS be used to study proton beam induced shockwaves?

Equation of state giving shockwave velocity:

20 pps qusucu

For tantalum c0 = 3414 m/s

Can ANSYS be used to study proton beam induced shockwaves?

Equation of state giving shockwave velocity:

20 pps qusucu

For tantalum c0 = 3414 m/s

Cf: ANSYS implicit wave propagation velocity :

smE

c /334516600

107.185 9

ANSYS benchmark study: same conditions as Alec Milne/FGES study i.e.ΔT = 1000 K

The y axis is radial deflection (metres)

Comparison between Alec Milne/FGES and ANSYS results

Alec Milne/ FGES

ANSYS

Amplitude of initial radial oscillation

100 μm 120 μm

Radial oscillation period

7.5 μs 8.3 μs

Mean expansion/ deformation

150 μm plastic

deformation

160 μm elastic

deformation

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

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

ANSYS benchmark study: same conditions as Alec Milne/FGES study - EXCEPT ΔT = 100 K (not 1000 K)

21

),(,,

TE

trzr

= 400 x 106 Pa

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:

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

Elastic shock waves in a candidate solid Ta neutrino factory target

Temperature jump after 1 ns pulse

(Initial temperature = 2000K )

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)

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?)

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

Elastic shock wave studies for draft ISOLDE proposal

Temperature jump after 2.4 μs pulse

(Initial temperature = 2000K )

Elastic shock wave studies for draft ISOLDE proposal

Temperature profile at centre of cylinder over 4 x 250 ns bunches

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

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)

Comparison between Nufact target and ISOLDE test

Temperature jump after 2.4 μs pulse

(Initial temperature = 2000K )

-1.00E+09 -5.00E+08 0.00E+00 5.00E+08 1.00E+09 1.50E+09

Maximum negative stress(r=0)

Shockwave oscillationamplitude (r=0)

Maximum stress at surface

Shockwave oscillationamplitude at surface

Stress (Pa)

ISOLDE test

Nufact target

Peak power density = 300 J/cc in both cases

Effect of pulse length on shockwave magnitude

-8.00E+08

-6.00E+08

-4.00E+08

-2.00E+08

0.00E+00

2.00E+08

4.00E+08

6.00E+08

8.00E+08

1.00E+09

1.20E+09

1.00E-08 1.00E-07 1.00E-06 1.00E-05Proton beam pulse length (s)

Str

ess

(Pa)

Maximum negative stress(r=0)

Shockwave oscillation amplitude (r=0)

Maximum stress at surface

Shockwave oscillation amplitude at surface