Applicability of Analytical Models for Applicability of Analytical Models for Predicting Predicting Hugoniot of Pre-Pressed Low-Hugoniot of Pre-Pressed Low-Density Compacts of Iron Nano-particlesDensity Compacts of Iron Nano-particles

Chengda Dai, Daniel Eakins, Naresh Thadhani

School of Materials Science & Engineering School of Materials Science & Engineering Georgia Institute of Technology, Atlanta GA30332Georgia Institute of Technology, Atlanta GA30332

EPNM-2008, May 5-9, Lisse, Netherlands

Supported by ONR/MURI under grant N00014-07-1-0740.

OUTLINEOUTLINE

• Motivation and Approach

• Current Analytical Models and their Applicability to

Low-density Powder Compacts

• Hugoniot Measurement Experimental Procedure

• Results of Measured Shock Hugoniot of Nano-Fe

• Correlation of Model Predictions with Measured

Shock Compressibility of Nano-Fe Powders

MOTIVATIONMOTIVATION

Fabrication of bulk materials via shock compaction of

powders requires reliable design of fixture geometry

Fixture design depends on availability of measured or

calculated Hugoniot of pressed powders

Shock Hugoniot of low-density micro-size powders can

be calculated using isobaric/isochoric models

Shock Hugoniot of nano-sized powdersnano-sized powders (either calculated

or measured) currently unavailable

APPROACHAPPROACH

(a) Examine applicability of McQueen’s isochoric

model and Wu-Jing’s isobaric model for describing

shock compression of micron-sized powders

(b) Measure shock Hugoniot of 25 nm-Fe powders

pre-pressed to 35% and 45% initial density; and

(c) Correlate model predictions with experimental

measurements on 25-nm Fe powders

CURRENT ANALYTICAL MODELSCURRENT ANALYTICAL MODELS

Isochoric Approach – constant volume (McQueen et al’s)

21

)(

21

21

00

000

00

0

VV

V

EEVP

VV

V

VV

VP HH

1970) McQueen, G. (R.

21

21

00

0

HH PVV

V

VVV

P

Specific internal energy for porous and solid assumed same (E00=E00)

Grüneisen parameter assumed identical for porous and solid material

CURRENT ANALYTICAL MODELSCURRENT ANALYTICAL MODELS

Isobaric Approach – Constant Pressure (Wu-Jing Model)

Specific internal energy assumed same for porous and solid material

- & P-dependent parameter (R) assumed identical for porous & solid

)/1)(2/(1

]/))(1(2/)[2/(

)/1)(2/(1

)2/1(

000

PPR

RVVRPVPVVR

PPR

VRV

E

CCEE

E

HH

) ,0(

)2/(1

))(2/( 000

CCE

HH

VVP

R

VVRVV

SK

PR

Correlation with Experiments: micro-Fe powderCorrelation with Experiments: micro-Fe powder

McQueen’s model shows correlation up to 60% TMD

Wu-Jing model shows correlation up to 43% TMD

• McQueen’s model correlates well only up to 0 = 1.66 (~60% TMD)

•Wu-Jing’s method provides correlation up to 0 = 2.33 (43% TMD)

• Wu-Jing model can be potentially employed to calculate Hugoniot of nanopowders (0 1+2/0 )

Correlation with Experiments: micro-Fe powderCorrelation with Experiments: micro-Fe powder

STARTING NANO IRON POWDERSTARTING NANO IRON POWDER

MONO-SIZED 25 nm bcc-IRON POWDER PARTICLESMONO-SIZED 25 nm bcc-IRON POWDER PARTICLES

HUGONIOT MEASUREMENTS ON NANO-IRON HUGONIOT MEASUREMENTS ON NANO-IRON

GAS-GUN IMPACT EXPERIMENTS GAS-GUN IMPACT EXPERIMENTS (STRESS & SHOCK VELOCITY (STRESS & SHOCK VELOCITY MEASUREMENTS) MEASUREMENTS)

Calculate: Particle Vel, Specific VolMeasure: Stress profile (σ(t)), Shock velocity (D)

t

iidt

R

tV

Adt 0

)(11)( D=hs/ (tA-tB) u= /(00D) /00 = D/(D-u)

50 mm Φ x 3 mm thick powder sampleInput PVDF gauge

Typical input and propagated stress tracesTypical input and propagated stress traces

35% TMD 45% TMD

Experimental data for ~25nm Fe (~35% and ~45% TMD)

Hugoniot for ~25nm-Fe powderHugoniot for ~25nm-Fe powder

Shock velocity extrapolated to ambient P: • 0.8 km/s for 35% TMD sample• 1.1 km/s for 45% TMD sample • close to measured sound speed values.

Transition Stress of Linear Segments: • ~2 GPa for 35% TMD and• ~6 GPa for 45% TMD

Shock and Particle Velocity (D-u) Stress and Particle Velocity (σx-u)

Measured Shock Velocity versus Stress (D-σ) Hugoniot for ~25nm-Fe powder

D-x calculated using jump

condition:

D=C0/2+½(C02+4S σxV00)½

Consistent with direct measurements, suggesting steady/pseudo-steady propagation through nano-powders

~35% TMD

~45% TMD

• Measured Hugoniot for ~25nm-Fe powder shows deviation from static curve

Wu-Jing Model Correlation with Wu-Jing Model Correlation with Measured Hugoniot of 25 nm FeMeasured Hugoniot of 25 nm Fe

~35% TMD ~45% TMD

Vi/Vo = (Voo/Vo) γ/(γ+2)

• Measured compression-to-expansion transition: Vi/V0 = 1.3 (for 35%) and = 1.08 (for 45% TMD) is same as obtained from calculation inflection

Inflection Volume

Correlation of Wu-Jing Model Prediction with Correlation of Wu-Jing Model Prediction with Experimentally Measured Hugoniot for 25 nm FeExperimentally Measured Hugoniot for 25 nm Fe

35% TMD (αo ≈ 2.86)

45% TMD (αo ≈ 2.22)

)/1)(2/(1

]/))(1(2/)[2/(

)/1)(2/(1

)2/1(

000

PPR

RVVRPVPVVR

PPR

VRV

E

CCEE

E

HH

) ,0(

)2/(1

))(2/( 000

CCE

HH

VVP

R

VVRVV

Wu-Jing Model with StrengthWu-Jing Model with Strength

Wu-Jing Model without StrengthWu-Jing Model without Strength

Wu-Jing model is ineffective in predicting Hugoniot of low-density nano-Fe in spite of its good correlation with micro-scale powders

CONCLUDING REMARKSCONCLUDING REMARKS

McQueen’s model is insufficient for Hugoniot prediction for

highly porous micro-scale powder (0 1+2/0).

Wu-Jing’s model capable of describing shock compression of

low-density micron-powder compacts, cannot describe

Hugoniot of nano-Fe powder

Wu-Jing’s and McQueen’s methods need to consider

characteristic properties of high surface area of nano- particles

to better predict Hugoniot of nano-particles

JOURNAL OF APPLIED PHYSICS 103, 093503 2008