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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)
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