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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)
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Courtesy: NAVSEA (ONR) presentation by Dr. Tom Moyer, 15th April, 2008
Underwater explosion phenomena
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Shock Wave
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Bubble Pulse
Courtesy: Snay et al. (1956)
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Bubble Collapse and Jetting
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Cavitation
Bulk Cavitation
Local Cavitation
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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
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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
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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
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• Rice and Walsh (1957)
• Al’tshuler et al. (1958)
• Bogdanov (1992) & Raybakov (1996)
• Nagayama et al. (2002)• Valid till 25 GPa
Us-up relationship
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• 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
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• 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
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• 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)
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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
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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
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Light plate limit Heavy plate limit
CBP VBP CBP VBP
Uniform
Exponential
FSI
CBP VBP
16
Analytical model
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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
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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
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Results
Validation: Numerical with Analytical
19
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Pressure history
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Comparison with existing theories :
β021
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Proposition of design curve for impulse transmission: Uniform Shock:
CR
22
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Proposition of design curve for impulse transmission: exponential shock
VBP case
23
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• 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
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25
RPPL model used in studying impact and shock problems
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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
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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
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Results
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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
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Results