Modeling Laser Material Strength Experiments Steve Pollaine David Petersen Lawrence Livermore National Laboratory 8th IWPCTM December 10-14, 2001 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 of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
Outline of poster •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
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 > 10 8 /s is unknown Relevant for impact of micrometeorites on space hardware Diagnostics VISAR X-ray diffraction Recovery Recovery Infer properties such as EOS, K, G, Y X-ray diffraction VISAR
Moderate shocks show both elastic and plastic waves Hugoniot Pressure Plastic flow and Pressure P- σ zz Hugoniot Plastic work-hardening elastic limit Elastic Elastic Elastic release Strain θ - ε zz Volume Pressure wave Pressure Plastic Elastic Distance
We use a material strength package in our code Newton’s law ρ = density ∂ ∂ 1 2 v= velocity ( ) + ( ) « ρ v = − r P − σ ∂ σ + σ + σ r rr rz rr zz P= hydrodynamic pressure ∂ z r σ = deviatoric stress ∂ ∂ 1 ( ) + « ρ v = − z P − σ ∂ σ + σ θ = hydrodynamic strain z zz rz rz ∂ r r ε = deviatoric strain K= bulk modulus Definition of strain G=shear modulus ∂ v ∂ v v EOS with strain « r z r θ = + + ∂ r ∂ z r « « « = − θ − P K P inelastic 1 ∂ v ∂ v v « r z r ε = − − inelastic 2 3 2 « « « σ = 2 G ε − σ + 2 σ ω + ( σ − σ ) ω rr rr rr rr rz zz rr ∂ ∂ r z r inelastic 2 « « « σ = 2 G ε − σ − 2 σ ω − ( σ − σ ) ω ∂ ∂ 1 v v v zz zz zz rz zz rr « z r r ε = 3 2 − − inelastic 2 « « « zz σ = 2 ε − σ + 2 ( σ − σ ) ω − 2 σ ω G ∂ ∂ z r r rz rz rz rz rr rz 1 ∂ v ∂ v 1 ∂ v ∂ v « r z r z ω = − ε = + rz ∂ ∂ 2 ∂ z ∂ r 2 z r
We use a von Mises yield criterion for the onset of plastic flow 4 ( ) 2 2 2 Deviatoric strain invariant J = σ + σ + σ + σ σ rr zz rz rr zz 3 ( ) 2 Effective pressure P = P − σ + σ ( σ − σ σ )/ 16 3 e rr zz rz rr zz When J > Y(P e ), the elastic limit is exceeded and plastic flow begins Y(P e ) Inviscid plastic flow (no viscosity) Elastic region Y 0 P e P min von Mises, Z. Angew. Math. U. Mech. 8 (1928), translated in UCRL Trans. 872
Uniaxial strain equations ∂ ( ) « ρ v = − P + σ z zz ∂ z ∂ v « z θ = ∂ z ∂ 2 v « z ε = zz ∂ 3 z « « « P = − K θ − P inelastic inelastic « « « σ = 2 ε − σ G zz zz zz 1 = − σ P P e 4 zz J = σ zz ( ) 4 Sound speed c 11 = K + G / ρ 3
Steinberg-Guinan Model ∂ ∂ 1 G P 1 G = + − − G P T ( , ) G 1 T T ( 300 ) 0 1 3 / ∂ η ∂ G P G 0 0 Y = Y f ( ε ) G P T ( , )/ G 0 p 0 n ( ) ≤ ε = + β ε + ε Y f ( ) Y 1 ( ) Y 0 p 0 p i max ρ ( ) 2 ( γ − − a 1 3 / ) 1 η η T = T exp 2 a ( 1 − ) , = 0 melt 0 η ρ 0 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 ( « , ) = { ε + ( ε )} ( , )/ Y Y T Y f G P T G T p A p 0 1 2 U Y C 2 − 1 « K T 2 ε = { exp[ ( 1 − ) ] + } p C kT Y Y 1 P T n ( ) ≤ ε = + β ε + ε Y f ( ) Y 1 ( ) Y A p A p i max ≤ Y Y T P D.J. Stenberg and C.M. Lund, J. Appl. Phys. 65 , 1528 (1989)
VISAR measures the surface velocity history Shield LiF window Etalon Laser Interface motion Interference Fiducial Al-6061 fringes timing marker •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
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 21382_fidu_lineout 1.0 0.8 Fiducial Particle speed (µm/ns) 0.6 Free surface 0.4 Plastic 0.2 LiF Elastic 0.0 Elastic Plastic -0.2 20 25 30 35 40 45 Time (ns) 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 Pull-back from spall 21384_fidu 1.0 Free surface 0.8 Particle speed (µm/ns) 0.6 Plastic 0.4 LiF 0.2 Elastic 0.0 -0.2 20 25 30 35 40 45 Time (ns)
We use VISAR data to determine the shear modulus, bulk modulus and yield strength Velocity = 2 P v P U ρ P e 2 P YU e e = = v e U ρ G e 0 Time t e = L 1 /u e + t 1 t p = L 2 /u p + t 2 4 K + G K 2 = ρ 2 3 = U U e p ρ 0 e
Shocks lose strength as they propagate + dP dP u c material s = − − 1 dx dx U shock rarefaction shock P X
The Steinberg-Guinan model by itself gives a spall time that is too late compared to the data Data Velocity Steinberg-Guinan µ m/ns Steinberg-Lund SG+Steinberg-Tipton failure model eps max = 0.25, ( ρ / ρ 0 ) min = 0.9665 Time (ns)
Steinberg-Tipton Failure Model Damage ranges from 0 to 1 Broken material: Y b < P, G b /G 0 = Y b /Y 0 {P,G,Y} = damage*{P 0 ,G 0 ,Y 0 } + (1-damage)*{P b ,G b ,Y b } RC f ∑ 2 s i max( , 0 ) > 1 ∆ i X fmax d zone i Damage = dt f 2 ∑ i 0 max( , 0 ) < 1 i fmax i 4 G 0 C = f i = {eps, ρ / ρ 0 − 1 , P, σ , ∆σ } s 3 ρ
Parameters • Steinberg-Guinan • p min = -30 kb • ρ / ρ 0 = 0.9665 • K= 940 kb • G 0 = 325 kb • Y = 3.335 kb • eps max = 2.0 • Steinberg-Tipton • ρ / ρ 0 − 1 = -.0335 • eps = .25 • R = 10 20 • Steinberg-Lund • Y = 1.5 kb • c 1 = .71 • c 2 = .12 • u k = .31 • y prl = 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 Shift of diffraction signal X-ray source Shocked Bragg Unshocked Bragg Compressed lattice Pressure source 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 320 kbar Si (111) 130 kbar Unshocked lattice 1.05 Lattice compression, d/d o 1.00 0.95 Relaxation 0.90 Shocked lattice 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)
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 scale 1 — Si responds uniaxially Shock direction — Cu shows plastic deformation Lattice spacing perpendicular to Lattice spacing shock (Laue) Hohlraum parallel to shock (Bragg) Crystal X-ray source [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 Shot 1 (28102219) Shot 2 (28102216) Static Bragg Static Laue Static Bragg Static Laue Streaked Bragg Streaked Laue 1.04 1.00 0.96 0.92 0.88 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
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