Underwater explosion (non-contact high-intensity and/or near-field)

induced shock loading of structures

-Nilanjan Mitra- (With due acknowledgements to my PhD student:

Ritwik Ghoshal)

Courtesy: NAVSEA (ONR) presentation by Dr. Tom Moyer, 15th April, 2008

Underwater explosion phenomena

Shock Wave

Bubble Pulse

Courtesy: Snay et al. (1956)

Bubble Collapse and Jetting

Cavitation

Bulk Cavitation

Local Cavitation

Reflected wave

Taylor (1941)

pR=2ps

ps

Acoustic(air &water)

Constant backpressure

Reflectedwave

Kambouchev et al. (2006)

pR=CRps

ps

Non-linear Compressible

(air )

Constant backpressureCR ˃2

Reflectedwave

Liu and Young (2008)

pR=2ps

ps

Acoustic(Water)

AcousticWater-backed

Reflectedwave

Peng et al. (2011)

pR=CRps

ps

Non-linearCompressible

(air )

Variable backpressure

Non-linearCompressible

(air )

CR ˃2

Shock Theories

7

Reflectedwave

ps

Non-linearCompressible

water

Variable backpressure

Non-linearCompressible

water

• Nonlinear compressible water both front and the back.

• Can capture intense shock events such as phase transition.

Present Theory (2012)

8Refer: Ghoshal and Mitra (2012), Journal of Applied Physics, 112(2), 024911

Equation of state (EOS)

• Ideal Gas pc Not considered

• Tait EOS Adiabatic, reversible

• Mie-Grüneisen Takes account pc and (MGEOS) pvib properly

• Polynomial Derived from MGEOS

Lattice configuration Thermal Vibration of ions

conductionelectron thermal

excitations

P>100 GpaT>104 K

• Rice and Walsh (1957)

• Al’tshuler et al. (1958)

• Bogdanov (1992) & Raybakov (1996)

• Nagayama et al. (2002)• Valid till 25 GPa

Us-up relationship

• Rice and Walsh (1957)

• Al’tshuler et al. (1958)

• Bogdanov (1992) & Raybakov (1996)

• Nagayama et al. (2002)

•Valid till 80 Gpa

• Shock compression may lead to formation of Ice VII.

•Break down of linear fit

Us-up relationship

• Rice and Walsh (1957)

• Al’tshuler et al. (1958)

• Bogdanov (1992) & Raybakov (1996)

• Nagayama et al. (2002)

Ice VII

Water

3

21

A

B

C D

T (K)

P (GPa)

A

BC

D

Particle Velocity (km/s)

Shoc

k Vel

ocity

(km

/s)

Us-up relationship

• Rice and Walsh (1957)

• Al’tshuler et al. (1958)

• Bogdanov (1992) & Raybakov (1996)

• Nagayama et al. (2002)

• Confirmed the formation of Ice VII.

• Pressure dependence of refractive index.

A

BC

D

Particle Velocity (km/s)

Shoc

k Vel

ocity

(km

/s)

Nagayama et al. (2002)

Rankine-Hugoniot Jump conditions

Us

p0u0ρ0e0

p1u1ρ1e1

Ps

P0

UR

p2u2=0ρ2e2

p1u1ρ1e1

PR

Ps

Incident Shock Reflected ShockConservation Equation

Mass

Energy

Momentum

Mie-Gruniessen EOS:

Analytical model

• Input Ps Output PR CR = PR / Ps

• Cubic Polynomial Of PRRoots :

• Complex Roots Neglected• CR > 2 Selected

1D Shock Reflection from a fixid rigid wall

Moving plate : Different shock profiles and backing conditions

Mass Conservation

Momentum Conservation

Free Standing Plate

Varying Back Pressure (VBP)

Constant BackPressure (CBP)

Uniform

Exponential

15

Analytical model

Light plate limit Heavy plate limit

CBP VBP CBP VBP

Uniform

Exponential

FSI

CBP VBP

16

Analytical model

Kinematic relations Momentum Energy

Numerical Analysis

Equation of state Artificial viscosity

Finite difference based VonNeumann-Richtmyer algorithm has been used for Shock capturing

Uniform

Exponential

mp VBP

CBPp

RLp m

ppA −=

17

Parameters used

Density of plate: 8000 kg/m3

Density of water: 1000 kg/m3

Parameters for Mie-Grüneisen EOS:

Segment I0<u<0.7

km/s

Segment II0.75<u<2

km/s

Segment III2.2<u<9

km/s

Fitting coefficient (S1) 2.116 1.68 1.185Bulk sound speed (c0) 1450 1879 2983

Courtesy: Bogdanov et al. (1992)Grüneisen parameter (Г0) = 0.28

18

Results

Validation: Numerical with Analytical

19

Pressure history

Comparison with existing theories :

β021

Proposition of design curve for impulse transmission: Uniform Shock:

CR

22

Proposition of design curve for impulse transmission: exponential shock

VBP case

23

• Core compression reduces back face-sheet velocity.

• Advantage due to FSI at the back isoverestimated.

Necessity of core compression model

RPPL (Rigid perfectly plastic locking) Model

24

• Face-sheets are assumed to be rigid.

• Elastic deformation of the core is neglected.

• Core becomes rigid after densification.

Extension of GM Theory for shocks to sandwich composite panels

Refer: Ghoshal and Mitra (2013), Journal of Applied Physics, Accepted

25

RPPL model used in studying impact and shock problems

Water-backed

Air-backed

Equation of motion Jump condition

Core compression model considering coupled effect of FSI at rear side of the plate

Assumption: Shock is arrested within the core

Necessary condition for plastic shock initiation within core

Derivation of Equation of motion and Jump conditions

Conservation of linear momentum,Lagrangian/material description,Small deformation

Integration over partial domains

separated by the plastic shock front discontinuity

yields equation of motion

Kinematic compatibilitycondition for discontinuity- Hadamard

Lagrangian/material description,Small deformation

Rate of change of linear momentum

Results

Energy conservation

Work done by incident pressure

(as per Fleck-Deshpande -- acoustic theory)

Rate of energy dissipation

Kinetic Energy rate

Work done by pressure on right side

Results