Modeling Laser Material Strength Experiments Steve Pollaine David - - PowerPoint PPT Presentation

• D. Kalantar, B. Remington, J. Belak, J. Colvin,
• M. Kumar, T. Lorenz, S. Weber

Lawrence Livermore National Laboratory

• J. Wark, A. Loveridge, A. Allen

University of Oxford

• M. Meyers

University of California, San Diego

This presentation was reviewed and released as UCRL-PRES-143513-REV-2. This work was performed under the auspices of the U.S. Department of Energy by University

• f California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

Modeling Laser Material Strength Experiments Steve Pollaine David Petersen Lawrence Livermore National Laboratory 8th IWPCTM December 10-14, 2001

• Material strength model
• Elastic-plastic flow
• Steinberg-Guinan and Steinberg-Lund models
• VISAR velocity measurement
• Experiment
• Model
• Diffraction
• Experiment
• Model
• Sample recovery
• Experiment
• Decay of shock strength
• Summary and future developments

Outline of poster

Recovery

The constituitive properties of metals is of general scientific interest

Laser experiments give us access to new regimes High pressures High strain rates How materials deform at strain rates > 108/s is unknown Relevant for impact of micrometeorites on space hardware Diagnostics VISAR X-ray diffraction Recovery Infer properties such as EOS, K, G, Y VISAR X-ray diffraction

Moderate shocks show both elastic and plastic waves

Elastic release

Strain Pressure

Elastic Plastic flow and work-hardening

Hugoniot Distance Pressure

Elastic Plastic

Pressure wave

Hugoniot elastic limit

Volume Pressure

Plastic Elastic

P-σzz θ-εzz

We use a material strength package in our code

ρ ∂ ∂ σ ∂ ∂ σ σ σ ρ ∂ ∂ σ ∂ ∂ σ σ « « v r P z r v z P r r

r rr rz rr zz z zz rz rz

= − −

( ) +

+ +

( )

= − −

( ) +

+ 1 2 1

« « « « θ ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ = + + = − −       = − −       = +       v r v z v r v r v z v r v z v r v r v z v r

r z r rr r z r zz z r r rz r z

1 3 2 1 3 2 1 2 « « « « « « ( ) « « « ( ) « « « ( ) P K P G G G v z v r

inelastic

rr rr rr inelastic rz zz rr zz zz zz inelastic rz zz rr rz rz rz inelastic rz rr rz r z

= − − = − + + − = − − − − = − + − − = − θ σ ε σ σ ω σ σ ω σ ε σ σ ω σ σ ω σ ε σ σ σ ω σ ω ω ∂ ∂ ∂ ∂ 2 2 2 2 2 2 2 1 2

2 2 2

     

ρ= density v= velocity P= hydrodynamic pressure σ= deviatoric stress θ= hydrodynamic strain ε= deviatoric strain K= bulk modulus G=shear modulus Definition of strain Newton’s law EOS with strain

We use a von Mises yield criterion for the onset of plastic flow

von Mises, Z. Angew. Math. U. Mech. 8 (1928), translated in UCRL Trans. 872

Deviatoric strain invariant J

rr zz rz rr zz

= + + +

( )

4 3

2 2 2

σ σ σ σ σ Effective pressure P P

e rr zz rz rr zz

= − +

( )

− σ σ σ σ σ ( )/

2 3

16

When J > Y(Pe), the elastic limit is exceeded and plastic flow begins Y(Pe) Pe Pmin Y0 Elastic region Inviscid plastic flow (no viscosity)

Uniaxial strain equations

ρ ∂ ∂ σ θ ∂ ∂ ε ∂ ∂ θ σ ε σ σ σ ρ « « « « « « « « « / v z P v z v z P K P G P P J K G

z zz z zz z zz zz zz inelastic e zz zz

inelastic

= − +

( )

= = = − − = − = − = = +

( )

2 3 2

1 4 4 3

Sound speed c11

Steinberg-Guinan Model

G P T G G G P P G G T T Y Y f G P T G Y f Y Y T T a

p p p i n melt a

( , ) ( ) ( ) ( , )/ ( ) ( ) exp ( ) ,

/ max ( / )

= + − −       = = + +

( ) ≤

= −

( )

=

− − 1 3 1 2 1 3

1 1 1 300 1 2 1 ∂ ∂ η ∂ ∂ ε ε β ε ε η η

η γ

ρ ρ0

D.J. Stenberg, S.G. Cochran and M. W. Guinan, J. Appl. Phys. 51, 1498 (1980) D.J. Steinberg, UCRL-MA-106439 (1991)

Steinberg-Lund Model

Y Y T Y f G P T G C U kT Y Y C Y Y f Y Y Y Y

T p A p p K T P T A p A p i n T P

= + = − + = + +

( ) ≤

≤

−

{ ( « , ) ( )} ( , )/ « { exp[ ( ) ] } ( ) ( )

max

ε ε ε ε β ε ε

1 2 2 1

1 2 1 1

D.J. Stenberg and C.M. Lund, J. Appl. Phys. 65, 1528 (1989)

VISAR measures the surface velocity history

• An optical laser pulse is reflected from the free surface of the foil and

injected into an interferometer

• The phase of the fringe is proportional to the velocity of the free surface
• Spatial resolution of the VISAR system provides data on the rear-

surface motion with and without the LiF window

Laser Shield Al-6061 LiF window Interface motion Etalon Fiducial timing marker Interference fringes

VISAR measurement of elastic-plastic wave breakout in Al-6061

195 µm Al-6061, LiF over half of the rear surface Omega shot #21382 - 19 J on target

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 20 25 30 35 40 45

21382_fidu_lineout Particle speed (µm/ns) Time (ns) LiF Free surface Elastic Plastic Fiducial Elastic Plastic Fiducial

The wave profile shows a pull-back at higher drive pressure

195 µm Al-6061, LiF over half of the rear surface Omega shot #21384 - 33 J on target

LiF Free surface

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 20 25 30 35 40 45

21384_fidu Particle speed (µm/ns) Time (ns) Plastic Elastic Pull-back from spall

We use VISAR data to determine the shear modulus, bulk modulus and yield strength

te = L1/ue + t1 tp = L2/up + t2

U K G

e 2 4 3

= + ρ U K

p e 2 = ρ

v P U YU G

e e e e

= = 2 ρ

Time Velocity

v P U

P P e

= 2 ρ

Shocks lose strength as they propagate

dP dx dP dx u c U

shock material s shock

  = −   + −      

rarefaction

1

P

X

Time (ns) Data Steinberg-Guinan Steinberg-Lund SG+Steinberg-Tipton failure model epsmax = 0.25, (ρ/ρ0)min= 0.9665 Velocity µm/ns

The Steinberg-Guinan model by itself gives a spall time that is too late compared to the data

Steinberg-Tipton Failure Model

Damage ranges from 0 to 1 Broken material: Yb < P, Gb/G0 = Yb/Y0 {P,G,Y} = damage*{P0,G0,Y0} + (1-damage)*{Pb,Gb,Yb}

d dt f f

i i i i

Damage RC X max( , ) max( , )

s zone

= > <       

∑ ∑

∆ 1 1

2 2

fmax fmax

i i

C G

s

= 4 3ρ

fi = {eps, ρ/ρ0−1, P, σ, ∆σ}

Parameters

• Steinberg-Guinan
• pmin = -30 kb
• ρ/ρ0 = 0.9665
• K= 940 kb
• G0 = 325 kb
• Y = 3.335 kb
• epsmax = 2.0
• Steinberg-Tipton
• ρ/ρ0−1 = -.0335
• eps = .25
• R = 1020
• Steinberg-Lund
• Y = 1.5 kb
• c1 = .71
• c2 = .12
• uk = .31
• yprl = 1.9 kb

Dynamic x-ray diffraction measures density and crystal structure

In situ x-ray diffraction allows us to probe the material state by

providing information on the lattice under compression

Technique applied on laser experiments at Nova and elsewhere (Janus,

Vulcan, Trident, OMEGA) and powder and gas gun facilities

Compressed lattice Shocked Bragg Unshocked Bragg X-ray source Pressure source Shift of diffraction signal

• Q. Johnson, A. Mitchell, R.N. Keeler, L. Evans, Phys Rev Lett 25, 1099 (1970)

J..S Wark, R.R. Whitlock, A.A. Hauer, J.E. Swain, P.J. Solone, Phys Rev B 40, 5705 (1989)

Diffraction from shock compressed Si has been demonstrated on Nova

Low intensity square laser pulse generates a single shock drive Displacement of the diffraction signal indicates a compression of the

lattice spacing

130 kbar 320 kbar 1.05 1.00 0.95 0.90 0.85

• 2.0
• 1.0

0.0 1.0

• 2.0
• 1.0

0.0 1.0 Time (ns rel. shock breakout) Time (ns rel. shock breakout) Lattice compression, d/do Unshocked lattice Shocked lattice Relaxation Si (111)

Diffraction from orthogonal lattice planes provides information on the transition to plasticity

Simultaneous measurements are made of compression of orthogonal

lattice planes

Shock compression above the HEL for Si and Cu show very different

behavior on the ns time scale1 — Si responds uniaxially — Cu shows plastic deformation

X-ray source Lattice spacing perpendicular to shock (Laue) Shock direction Hohlraum Crystal Lattice spacing parallel to shock (Bragg)

[1] A. Loveridge et al, "Anomalous elastic response of silicon to uniaxial shock compression on nanosecond timescales", Phys. Rev. Letters 86, 2349 (2001)

Simultaneous measurements of orthogonal planes indicate Si responds uniaxially on a ns time scale

Si shock compressed along (400); probed along (400), (040) P = 115-135 kbar; HEL = 84 kbar, 40 µm thick Si Simultaneous measurements of Bragg and Laue diffraction

0.88 0.92 0.96 1.00 1.04 Shot 1 (28102219) Shot 2 (28102216) Static Bragg Streaked Bragg Static Laue Static Bragg Streaked Laue Static Laue 2.0 5.0 8.0 2.0 5.0 8.0 Time (ns) Time (ns)

1-D compression in Si is due to high Peierls barrier

• 100

100 200 300 1000 1500 2000

Pressure (kb) Longitudinal stiffness C11 (kb) MD simulation C11 = K+4/3 G = pr+1650

Molecular dynamic simulations show that the Si longitudinal stiffness increases with pressure

Simulation done by D. J. Roundy

We have recovered samples to study the residual effects due to these high strain rate laser experiments

Single crystal Cu samples were shocked by direct laser irradiation and

captured in a foam-filled cavity

Preliminary tests done at OMEGA; shock pressure is >1 Mbar, decays

to ~50 kbar at the rear surface

Side Back

1 mm Drive Beam Recovery tube 50 mg/cm3 foam Cu crystal

Recovery

We see spall on a Cu sample driven by Janus

Summary and future work

• VISAR provides free surface velocity history
• Gives shear modulus, bulk modulus and yield strength
• Gives information on fracture model and spall
• X-ray diffraction provides information about lattice deformation
• Future work
• Correlate VISAR with x-ray diffraction
• Relate VISAR with post-shock recovery and residual deformation
• f structure