SHOCK WAVES IN THE PHYSICS OF CONDENSED MATTER

G.I. Kanel, S.V. Razorenov, V.E. FortovJoint Institute for High Temperatures of Russian Academy of Sciences

Institute of Problems of Chemical Physics of Russian Academy of Sciences

• The method• Polymorphic and phase transformations• High-rate plastic deformation and fracture• Approaching the ideal strength• Failure waves• Behavior of hard brittle materials

APPEARANCE OF MATERIAL PROPERTIES IN A FREE SURFACE VELOCITY HISTORY

Available loading conditions:

•Peak stresses: 0.1 −

100 GPa•Load durations: 10 ns −10 μs•Time resolution of the measurements:

< 1 ns

Hugoniot elastic limit: HEL = ρ0 cl ufsHEL/2

Yield stress: Y = HEL (1-2ν)/(1-ν)

Spall strength: σsp = ρ0 cb (Δufs + δ)/2

Phase transition pressure:

pα→ε

= ρ0 D1 ufs1 /2

Sample

VISAR ufs (t)

Flyer plate

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2Steel 40Kh

Spallsignal

Δufs

Polymorphous transformation

Hugoniot elastic limit (HEL)

Free

Sur

face

Vel

ocity

, km

/s

Time (μsec)

DIAGNOSTICS: measurements of pressure, tests at elevated

temperatures

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

Armco iron 2.46 mm

590oC

480oC

20oC

Free

Sur

face

Vel

ocity

, km

/sTime, μs

Extremely high transformation rate at shock-wave compression;Sub-microsecond polymorphic transformations have been studied for iron and steels, titanium, tin, graphite, boron nitride, …

0 1 2 3 40

10

20

30

Steel 35Kh3NM

Armco-iron

Com

pres

sive

Stre

ss, G

Pa

Time, μs

POLYMORPHIC TRANSFORMATIONS

Highly Oriented Highly Oriented PyroliticPyrolitic GraphiteGraphite (HOPG)(HOPG)

•

The increase of mosaic spread decreases the transformation pressure.•

Transformation of HOPG occurs with acceleration whereas pressed graphite shows acceleration in initial stage and deceleration at the end.

•

The maximum transformation rate is 2.5×107 s-1 for HOPG and 8 ×106 s-1 for pressed graphite.

0,0 0,5 1,00,0

0,5

1,0

1,5

HOPG Pressed2.17 g/cm3,p3 = 0.85 - 1

MS = 1.7o

p3 = 0.74

2.25 gr/cm3

MS < 0.2o

p3 = 0.68Pa

rticl

e ve

loci

ty, k

m,s

Time, μs

GraphitesGraphites of various grain sizeof various grain size

0,00 0,25 0,50 0,75 1,000,0

0,5

1,0

1,5

0,0 0,2 0,41,2

1,4

1,6

2.03-2.08 g/cm3

20-50 μm

Expanded,5−30 nm

0.25 - 4 μm

Parti

cle

velo

city

, km

/s

Time, μs

• Fine-grained graphites demonstrate higher lower average transformation rate but faster stress relaxation in the initial stage.

• This may be interpreted as faster nucleation in the material with large intergranular surface.

Expanded graphite

TRANSFORMATION GRAPHITE-DIAMOND UNDER SHOCK COMPRESSION

LiF

Win

dow

LiF

Win

dow

Gra

phite

Gra

phite

II T

Dependence of the transformation pressure and rate on the load direction confirms the martensitic nature of the transformation.

0,0 0,5 1,00,0

0,5

1,0

1,5

II

2.185 g/cm3, 2.87 mmT

2.169 g/cm3, 4.25 mm

Highly oriented graphite,pressedFr

ee S

urfa

ce V

eloc

ity, k

m/s

Time, μs

Temperature dependences of the transition pressureTemperature dependences of the transition pressure

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

ρ0=2.08 g/cm3,5 vol % rhom. ph.

ρ0=2.148 g/cm3,25-28 vol % rhom. ph.

Exp.

gra

phite

-dia

mon

d co

exist

ence

(with

cat

alys

ts)

Cubic diamond

cubic + hexagonal+ graphite

Graphite

Tem

pera

ture

, 0 C

Pressure, GPa

transparentdiamond

The two kinds of graphite with different transformation pressures demonstrate the same slope of their ptr (T) dependences.Extrapolated ptr (T) dependences intersect the graphite-diamond coexisting line at the lower temperature of formation of quenchable cubic diamond.

Shape memory materials

MiA

iA

kM

kM

d

σM

εεεεεε

Stre

ss

Temperature

-200 -100 0 100 200 3000

200

400

600

800

Ti - 49 at.% Ni

Ti - 51 at.% Ni

Stre

ss, M

Pa

Temperature, oC

Common plasticity

Plasticity due to martensitic transformationElasticity

Y ϕ

C

BA

Stre

ss

Strain

Cooling

Heating

DEFORMATION OF THE SHAPE MEMORY MATERIALS

0 100 200 3000

100

200

300

400

-78oC2.4 mm

30oC2.67 mm

140oC2.82 mm

Ni50,6-Ti49,4deformed

Free

Sur

face

Vel

ocity

, m/s

Time, ns

-200 -100 0 100 200 3000

200

400

600

800

1000

0

1

2

3

4

Ni50,6-Ti49,4deformed

Deformed

Ni48.9-Ti51.1

Ti - 49 at.% Ni statics

Ti - 51 at.% Ni, statics

Yiel

d St

ress

, MPa

Temperature, oC

Hug

onio

t ela

stic

lim

it, G

Pa

0 100 200 3000

100

200

300

400

Ni48.9-Ti51.1 2.75 mm20oCFr

ee S

urfa

ce V

eloc

ity, m

/s

Time, ns

Plasticity due to martensitic transformation

Elasticdeformation

B

Aσϕ

Stre

ss

Strain

DYNAMIC STRENGTH

SPALL PHENOMENA UNDER SHOCK LOADING

Spalling is the process of internal rupture of a body due to tensile stresses generated as a result of a compression pulse reflected from the free surface.

σx

σx

Free surface

x

Free surface

x

MANIFESTATIONS OF STRUCTURAL FACTORS AT SPALLING

• The spall strength is by ~15% less in the case of loading in the lateral direction;

• Faster decay of the velocity oscillations correlates with a more highly developed fracture surface;

0 1 2 3 4 5 60

100

200

300

400

500Rolled steel 35Kh3NM

Transversal direction

Rolling direction

Free

Sur

face

Vel

ocity

, m/s

Time, μs0 100 200 300 400

0

200

400

600

Annealed

"As-received” (quenched and aged)

AS

D16T (Al 2024)

Free

Sur

face

Vel

ocity

, m/s

Time, ns

• The “as received” alloy demonstrates high spall strength and relatively slow fracture process;

• Annealing resulted in decrease of the spall strength and fasten the fracture process;

Polycrystalline metals and metal single crystals

Highly homogeneous single crystals are free of potential fracture nucleation sites and demonstrate much higher spall strength than that of polycrystalline material.

0.0 0.1 0.2 0.3 0.4 0.50

200

400

600 Polycrystalline aluminum AD1

Al single crystal

Free

Sur

face

Vel

ocity

, m/s

Time, μs

Intragranular and intergranular fracture of copper

SPALL FRACTURE OVER WIDE RANGE OF THE LOAD DURATION

0 100 200 3000

200

400

600 single crystals

Cu

Free

Sur

face

Vel

ocity

, m/s

Time, ns

The load duration is varied from ~10 ns to ~10 μsThe resistance to spall fracture grows with increasing the strain rate

0 1 2 3 40

250

500

750

Aluminum AD1

Free

Sur

face

Vel

ocity

, m/s

Time, μs

T1<T2<T3<T4

T4

T3 T2 T1

Density

Pre

ssur

e Gas

?

Vaporization

Melting

Sol

id

Liqu

id

P > 0

P < 0

0

• The pressure of a gas cannot be negative.• Unlike gases, liquids and solids have a finite density at zero pressure

due to attractive intermolecular interactions.• Stretching a liquid or a solid means applying a negative pressure to it.• The stretched states are bounded by the spinodal.

( )kTEJJ a−= exp0 ( )

21

0

3

ln316)( ⎥

⎦

⎤⎢⎣

⎡Δ

−=tVJkT

Tpcπγ

104 105 106 107σ sp

/ σ id

WaterPMMA

Epoxy

0.04

0.06

0.08

0.50.4

0.3

0.2

0.1

CuAl

Fe

Mo

.V/V

0, s-1

APPROACHING THE IDEAL STRENGTH

• As much as 30 % of ideal strength is reached at load duration of a nanosecond range.

0,6 0,8 1,0 1,2-60

-40

-20

0

20

Water

PMMA

Mo

Fe

Cu

Al

Tension Compression

Pres

sure

, GPa

ρ/ρ0

NUCLEATION OF FRACTURE. Molecular-dynamic simulations.

Thermal effects

SHOCK-WAVE LOADING OF ALUMINUM SINGLE CRYSTALS AT ELEVATED TEMPERATURES

• High spall strength is maintained up to temperatures just by 10°

below the melting point;

• The Hugoniot elastic limit unexpectedly grows with increasing the temperature;• The velocity pullback (spall strength) decreases whereas the HEL increases with

heating;• A strong rate sensitivity results in a strong decay of the elastic precursor wave.

0.0 0.1 0.2 0.30

100

200

300

400

500

600

700

Sample thickness 2.85 mm.

407oC20oC

650oC

Free

Sur

face

Vel

ocity

, m/s

Time, μs0 20 40 60 80

0

200

400

600

800

260μm, 622oC

425 μm, 20oCFree

Sur

face

Vel

ocity

, m/s

Time, ns

SPALL STRENGTH OF SINGLE CRYSTALS AND POLYCRYSTALLINE METALS AT MELTING

• The strength of polycrystalline metals drops when the material begins to melt whereas single crystals maintain a high resistance to spall fracture when melting should start;

• In polycrystalline solids melting may start along grain boundaries at temperatures below the melting temperature of the crystal: pre-melting phenomenon;

• Superheated solid states were realized in the crystals under tension

0 100 200 300 400 500 600 7000

1

2

3

4

Zn (0001)

3.106s-1

5.105s-1

Al

Spal

l Stre

ngth

, GPa

Temperature, oC0,4 0,6 0,8 1,0

0,0

0,5

1,0

1,5

MgAl

Spal

l Stre

ngth

, GPa

- Magnesium Mg95- Aluminum AD1

Homologous Temperature T/Tm

ANOMALOUS GROWTH OF THE DYNAMIC YIELD STRENGTH OF ALUMINUM SINGLE

CRYSTALS WITH INCREASING THE TEMPERATURE

• Dynamic yield strength increases linearly with increasing temperature;• Yield strength near the melting temperature exceeds its value at room temperature by a

factor of four and the ratio of the yield strength to the shear modulus increases by an order of magnitude (at low strain rates this ratio decreases to half of this value as the temperature is increased to 900K)

• Under conditions of shock deformation of aluminum the dislocation drag is determined by thermal oscillations of atoms.

0 100 200 300 400 500 600 7000.0

0.2

0.4

0.6

0.8

1.0

Y

Valleys

Peaks

Valleys

Peaks

HEL

Tm

Stre

ss, G

Pa

Temperature, oC

Shoc

k W

aves

Quasistatic, Hopkinson bars

Phonon drag controlled flow

Thermally activated flow

T2 > T1

T1

Flow

Stre

ss

Strain Rate

Anomalous thermal hardening

0.1 1 100.0

0.2

0.4

0.6

5 x 103 s-1

1.5 x 104 s-1

1.5 x 105 s-1

σHEL = 0.2 - 0.3 lg(x)

AD1

1060 Al

1050 Al

σ HEL

, GPa

Distance, mm

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

0.00 0.01 0.02 0.030

40

80

120

5.4 mm0.57

10.0 mm

2.2 mm

AD1 20oC

Free

Sur

face

Vel

ocity

, m/s

Time, μs

2.180.57

t/h, μs/mm

ufs, m/s

0.1 1 100.0

0.5

1.0

1.5

2.0

2.5

3.0

2.4 x 103 s-1

4.8 x 104 s-1

0.655-0.42*log(x)

20 oC

Single crystals622oC

~600 oC

AD1σ H

EL, G

Pa

Distance, mm

γρσ&l

HEL cdh

d02

1−=

SHOCK WAVES IN TI-6-22-22S SAMPLES AT NORMAL AND ELEVATED TEMPERATURES

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

10 mm 2.24mm

20oC

Free

Sur

face

Vel

ocity

, m/c

Time, μs

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

10 mm, 620oC

2.24 mm, 600oC

Free

sur

face

vel

ocity

, m/c

Time, μs

10-6 10-4 10-2 100 102 104 1060.0

0.5

1.0

1.5

2.0

1-D strain (Shock)

Uniaxial stress

Yiel

d S

tress

, Y0.

2, G

PaStrain Rate, s-1

General dependence and dominating mechanism of dislocation motion are maintained at highest strain rates

BEHAVIOR OF BRITTLE MATERIALS

COMPRESSIVE FRACTURE OF BRITTLE MATERIALS

Open cracks may appear only under tensile stresses.Even when applied stress is wholly compressive, the local stress may become tensile at certain points at pre-existing crack tip.

Tension

Tension

Compression

Compression

Tensilecracks

Wing crack

Griffith’s fracture criterion: the fracture occurs when the highest local tensile stress at the longest crack of the most dangerous orientation reaches a critical value.

SHOCK COMPRESSION AND SPALL FRACTURE OF ALUMINA CERAMIC

• Ceramic demonstrates remarkable spall strength when the spallation occurs after elastic compression or after compression at insignificant plastic deformation.

• The waveforms do not indicate any signature of compressive fracture

0.00 0.25 0.50 0.75 1.000

200

400

600

800

1000

2.2mm4.5mm

δ

δ3.46 g/cm3

Al2O3

Free

Sur

face

Vel

ocity

, m/s

Time, μ s

BRITTLE-DUCTILE TRANSITION AT HIGH PRESSURE

• Below 0.2 GPa of the confining pressure the ultimate compressive strength increases rapidly with pressure growth.

• The brittle-ductile transition occurs within 0.2−0.3 GPa pressure range.

• Above 0.3 GPa the ultimate strength is growing much slower.

• Ceramic exhibits an increasing ductility when the confining pressure arises above the brittle-ductile transition.

• At a low confining pressure, the failure occurred by macroscopic shear on fault planes, while the rest material contained numerous axial microcracks.

• At a confining pressure > 0.5 Gpa, the dominant deformation mechanism became intracrystalline slip by dislocation motion.

0 5 100

1

2

3

Brittle-Ductile transition

Ductile response

Brittle response

1.0

0.5

0.3 GPa0.2

0.1GPa0

σ1-

σ3(

GP

a)

ε 1(%)

BeO ceramic

H.C. Heard, C.F. Cline, J. M at. Sc., 15, 1889 (1980)

EXPERIMENTS WITH SPECIMENS CONFINED BY SHRINK-FIT SLEEVE

The confining stress produced by the sleeve is controlled by the misfit δ

value and the yield stress of the sleeve material.

Ø

Ø -δ

FITTING OF THE DISK SAMPLE INTO THE COMPRESSIVE RING

(Zaretsky et al., 2002)

Guide bush

Sample discSteel ring

Sample disc

Piston

20o C

600o C

DUCTILE AND BRITTLE RESPONSE OF CERAMICS UNDER UNIAXIAL SHOCK COMPRESSION

Boron carbide exhibit brittle response whereas the behavior of alumina is ductile

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

100

200

300

400

500

HEL

Fre

e S

urfa

ce V

eloc

ity, m

/s

Time, μ s

0.00 0.02 0.04 0.06 0.08 0.10-10

0

10

20

30Al2O3

UnconfinedConfined

uc -

uu

0.0 0.2 0.4 0.6

0

200

400

600

800

1000

1200

B4C

Free sample

Pre-stressed

Fre

e S

urfa

ce V

eloc

ity, m

/sTime, μ s

Compressive fracture of the B4 C ceramic Computer simulations

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

14

σcInterparticle friction

σf

Failure threshold Uniaxial compression

B4C

Hydrostat

Lankford

Meyer

Wilkins - failed

Wilkins - initial

σ1

σ2=σ3

σ 1=σ 2

(1-ν)/ν

0.0 0.2 0.4 0.6 0.80

200

400

600

800

Experiment

SimulationB4C, 8 mm

Plate impact (Al 2mm, 1.9 km/s), LiF window

Par

ticle

Vel

ocity

, m/s

Time, μs

0 5 10 15 20 250

5

10

15

20

25

Elastic unloading

Fracture under compression

Interparticle friction

Inter

partic

le fric

tion

Frac

ture

Thr

esho

ld

HEL

B4C

Hydrosta

t

σ1

σ2=σ3

FAILURE WAVES IN SHOCK-COMPRESSED GLASSES

MEASUREMENTS OF DYNAMIC TENSILE STRENGTH OF A GLASS

The reverberation time trev of an elastic compression pulse is less than the expected one.The surface layer obviously has been damaged by cracks.

0.0 0.5 1.0 1.5 2.0 2.50

250

500

750

1000

Simulation

Experiment

Free

Sur

face

Vel

ocity

(m/s

)

Time (μs)

Glass plate

texpected

tr

t0

Tensile pulse

Impa

ctor

Unloading

Compression wave

t

x

Cra

cks

Kanel, G.I., Molodets, A.M., and Dremin, A.N., Fizika Goren. Vzriva, 13(6), p. 905 (1977)

FAILURE WAVE IN SHOCK-COMPRESSED GLASS

• The failure wave is a network of cracks that are nucleated on the surface and propagate with subsonic speed into the stressed body;

• Investigation of failure wave in shock-compressed glasses may provide information about the mechanisms and general rules of nucleation, growth and interactions of multiple cracks and lead to a better understanding of experiments on brittle ceramics and rocks.

Dis

tanc

e

Time

Re-re

flecti

onUnloading

Elas

tic sh

ock

Δtr

Glass Plate

GlassSurface

Failure wave

Base plate

Incipient microcracks always exist on the glass surface.

TRANSFORMATION OF SHOCK COMPRESSION PULSES IN A PILE OF

SODA-LIME GLASS PLATES

• Superposition of failure waves forms the compaction wave: elastic wave transforms into an elastic-plastic wave;

• Magnitude of the precursor wave is the failure threshold;• The pile may be considered as a model of polycrystalline brittle

material.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

VISAR VISAR

Free

Sur

face

Vel

ocity

, km

/s

Time, μs

tf

h hDc

tD

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7 σfσc

Yield s

tress

Com

minu

ted

state

s

Failu

re th

resh

old

Pressu

re (m

ean s

tress

)

σx

σy, GPa

COMPUTER SIMULATION OF THE FAILURE WAVE: COMBUSTION-LIKE MODEL

Copper impactor Glass plate

Propagation of fracture:

When the material becomes failed the stress deviator relaxes to σc

(Y. Partom. Int. J. Impact Engng. 21(9), 791, 1998)

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

ufs

Stress at impactor/sample interface

Stre

ss, G

Pa

Time, μs

0.0

0.2

0.4

0.6

0.8

1.0

Fre

e Su

rface

Vel

ocity

, km

/s

Each surface is a fracture nucleation site.The model reproduces the decrease of magnitude of leading elastic wave at each interface, the free surface velocity history, and the stress oscillations on input glass surface.

COMPUTER SIMULATION OF THE FAILURE WAVE: COMBUSTION-LIKE MODEL

Copper impactor Glass pile

Computer simulations with accounting for

anomalous growth of the Poisson’s ratio

4 6 8 10 12 140

1

2

3

4

5

6

7

1.5Gla

ss

Cop

per b

ase

plat

e

1.2 μs0.9 μs0.6 μs0.3 μs

Stre

ss, G

Pa

Distance, mm

1,5 2,0 2,5 3,00,0

0,2

0,4

0,6

0,8

1,0

Pile 8 x 1.2 mm

ExperimentSimulation

Free

Sur

face

Vel

ocity

, km

/s

Time μs0,0 0,5 1,0 1,5 2,0 2,5

0,2

0,3

0,4

0,5

GPa

ν = 0.2617 + 0.01428 (σx-p)3

ν

σx-p,

0,25

0,30

0,35

0,40

0,45

0,50

0 5 10 150

2

4

6

8

10

p=46.2*ε

Failure Wave Range

p

σx

Stre

ss, P

ress

ure,

GPa

Strain ε = 1-V/V0, %

ν

ν=(3-ρ0aσ

2/46.2)/(3+ρ0aσ

2/46.2)

ν

PRE-STRESS EFFECT ON THE FAILURE WAVE

0.0 0.5 1.0 1.50

200

400

600

8005.5 GPa

Pre-stressed,210 MPa

Free

K8 crown glass3.51 mm5.7 GPa

t=1.17 μs - expected reflection from the impact surfaceFr

ee S

urfa

ce V

eloc

ity, m

/s

Time, μs0.0 0.5 1.0 1.5

0

200

400

600

800

t=1.19 μs - expected reflection from the impact surface

Pre-stressed160 MPa

Free

K14 crown glass3.5 mm, 5.2 GPa

Velo

city

, m/s

Time, μs

Failure wave is stopped by unloading when the stress decreases down to the failure thresholdThe measurements clearly demonstrate the influence of pre-stressing on the distance of the failure propagation which appears in increased reverberation time. The effect is larger in the case of K14 glass which is characterized higher failure threshold.

Influence of a lateral pre-stressing on the compressive failure threshold

Ybr

Δ HEL (brittle)

π

Compr

essiv

e fa

ilure

thre

shold

Yd

σ1

σ2=σ3

σ1=σ2(ν-1)/ν

Uniaxial compression

Hydrostat

Sensitivity of the threshold value to small variations of transversal stress (Zaretsky et al., 2002):

( )( )( )ν

ννπ

σ21

231−

−−=

dd f

For glasses expected πσ dd f ≈

3.5

Upper estimation of real pre-stress effect Δσf on the failure threshold σf

22sffs

lf

ttuc

Δ−Δ=Δ

&ρσ

For K8 crown glass at 215 MPa radial confinement stress Δσf = 140 MPa, = 0.65

For K14 crown glass at 160 MPa radial confinement stress Δσf = 400 MPa, = 2.5

Obviously, there are some physical phenomena which we did not account for our analysis.

πσ fΔ

πσ fΔ

NUCLEATION OF CRACKS UNDER COMPRESSION

Tension

Tension

Compression

Compression

Tensilecracks

Wing crack

In the “wing crack” model, generation of local areas of tension is accompanied with appearance of local areas of excess compression.As a result, conditions for the irreversible densification are created.

GLASS AFTER SHOCK COMPRESSION

PMMAGlass

Sapphire

The strong irregular oscillations and low reproducibility of the waveforms are a consequence of intrinsic heterogeneity of its inelastic deformation. The precursor waveforms with spikes are associated with accelerating stress relaxation behind the precursor front.

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.25

0.50

0.75

1.00

No strength

High strength

Simulation

1

Sapphire c-cut5 mm, LiF window,uimp=1.8 km/s

Parti

cle

velo

city

, km

/s

Time, μs

0.00 0.25 0.50 0.750

50

100

150

200

250

Zn single crystal, plane:

2.25 mm

1.84 mm

_

(1010)

Free

Sur

face

Vel

ocity

, m/s

Time, μs

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

250

Titanium (Zaretsky, 2004)

Free

Sur

face

Vel

ocity

, m/s

Time, μs

c-cut sapphire at intermediate impact stress

0.0 0.2 0.4 0.6 0.80

400

800

1200

Hot pressed alumina 96% purity,3.93 - 3.94 g/cm3,6 mm thickness

HEL=7 GPa

Free

Sur

face

Vel

ocity

, m/s

Time, μs

• Comparison of the measured and simulated waveforms indicates negligible tensile strength after deformation in the plastic wave.

• At shock compression below the HEL, sapphire demonstrates the highest values of spall strength, which grow with shortening of the load duration.

• There is also a trend for the spall strength to decrease with increasing peak stress.

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.25

0.50

0.75

1.00

No strength

High strength

Simulation

1

Sapphire c-cut5 mm, LiF window,uimp=1.8 km/s

Par

ticle

vel

ocity

, km

/s

Time, μs

Spall strength of c-cut sapphire

0.0 0.1 0.2 0.3 0.4 0.50.00

0.25

0.50

0.75

1.00

10.4 GPa

σpeak = 23 GPa, σspall > 20 GPa

σpeak = 20.6 GPa, σspall = 4.2 GPa

σpeak = 18.2 GPa,σspall = 8.9 GPa

Free surface

Water window

Par

ticle

vel

ocity

, km

/s

Time, μs

• Irregular oscillations in the waveforms are of higher frequencies and lower amplitudes than those recorded for other orientations.

• The data are very reproducible and much less noisy than the waveforms of most other orientations.

• The rise time of the plastic wave at the intermediate peak stress is about 5 ns.• The stress at the front of the elastic precursor at intermediate impact velocity

is less than for a lower impact velocity.

s-cut and m-cut sapphire

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

uimp=5.2 km/s

uimp=1.2 km/s

uimp=1.8 km/s

Sapphire s-cut,5 mm, LiF window

Parti

cle

velo

city

, km

/s

Time, μs0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.25

0.50

0.75

1.00

c-cut

s-cut

m-cut

Sapphire 5 mmuimp=1.8 km/s,LiF window

Parti

cle

velo

city

, km

/s

Time, μs

Thank you for your attention!