Underwater explosion (non-contact high-intensity and/or near-field) induced shock loading of structures

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1 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)

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

3 Shock Wave

4 Bubble Pulse Courtesy: Snay et al. (1956)

5 Bubble Collapse and Jetting

6 Cavitation Bulk Cavitation Local Cavitation

7 Shock Theories Reflected wave p R =2p s Acoustic (air &water) p s Constant back pressure Reflected wave p R =C R p s C R 2 Non-linear Compressible (air ) Taylor (1941) p s Constant back pressure Kambouchev et al. (2006) Reflected wave p R =2p s Acoustic (Water) p s Acoustic Water-backed Reflected wave p R =C R p s C R 2 Non-linear Compressible (air ) p s Variable back pressure Non-linear Compressible (air ) Liu and Young (2008) Peng et al. (2011) 7

8 Present Theory (2012) Reflected wave Non-linear Compressible water p s Variable back pressure Non-linear Compressible water Nonlinear compressible water both front and the back. Can capture intense shock events such as phase transition. Refer: Ghoshal and Mitra (2012), Journal of Applied Physics, 112(2),

9 Equation of state (EOS) Lattice configuration Thermal Vibration of ions conduction electron thermal excitations Ideal Gas p c Not considered Tait EOS Adiabatic, reversible P>100 Gpa T>10 4 K Mie-Grüneisen Takes account p c and (MGEOS) p vib properly Polynomial Derived from MGEOS

10 Us-up relationship Rice and Walsh (1957) Al tshuler et al. (1958) Bogdanov (1992) & Raybakov (1996) Nagayama et al. (2002) Valid till 25 GPa

11 Us-up relationship Rice and Walsh (1957) Al tshuler et al. (1958) Bogdanov (1992) & Raybakov (1996) Valid till 80 Gpa Shock compression may lead to formation of Ice VII. Break down of linear fit Nagayama et al. (2002)

12 Us-up relationship T (K) 3 Rice and Walsh (1957) Al tshuler et al. (1958) Water 1 A B 2 C D Ice VII Bogdanov (1992) & Raybakov (1996) P (GPa) Nagayama et al. (2002) Shock Velocity (km/s) A B C D Particle Velocity (km/s)

13 Rice and Walsh (1957) Al tshuler et al. (1958) Confirmed the formation of Ice VII. Pressure dependence of refractive index. Bogdanov (1992) & Raybakov (1996) Nagayama et al. (2002) Shock Velocity (km/s) A B C D Nagayama et al. (2002) Particle Velocity (km/s)

Energy Reflected Shock p 1 u 1 ρ 1 e 1 P s P R U R p 2 u 2 =0 ρ 2 e 2 Mie-Gruniessen EOS: Input

14 Analytical model 1D Shock Reflection from a fixid rigid wall Rankine-Hugoniot Jump conditions P s p 1 u 1 ρ 1 e 1 Incident Shock U s p 0 u 0 ρ 0 e 0 P 0 Conservation Equation Mass Momentum Energy Reflected Shock p 1 u 1 ρ 1 e 1 P s P R U R p 2 u 2 =0 ρ 2 e 2 Mie-Gruniessen EOS: Input P s Output P R C R = P R / P s Cubic Polynomial Of P R Roots : C R > 2 Selected Complex Roots Neglected

15 Analytical model Moving plate : Different shock profiles and backing conditions Uniform Varying Back Pressure (VBP) Exponential Constant Back Pressure (CBP) Mass Conservation Free Standing Plate Momentum Conservation 15

16 Analytical model Light plate limit Heavy plate limit CBP VBP CBP VBP Uniform Exponential CBP VBP FSI 16

17 Numerical Analysis Kinematic relations Momentum Energy Equation of state Artificial viscosity Uniform m p VBP Exponential pl pr A CBP p = mp Finite difference based VonNeumann-Richtmyer algorithm has been 17 used for Shock capturing

18 Parameters used Density of plate: 8000 kg/m 3 Density of water: 1000 kg/m 3 Parameters for Mie-Grüneisen EOS: Segment I 0<u<0.7 km/s Segment II 0.75<u<2 km/s Segment III 2.2<u<9 km/s Fitting coefficient (S 1 ) Bulk sound speed (c 0 ) Courtesy: Bogdanov et al. (1992) Grüneisen parameter (Г 0 ) =

19 Results Validation: Numerical with Analytical 19

20 Pressure history

21 Comparison with existing theories : β 0 21

22 Proposition of design curve for impulse transmission: Uniform Shock: C R 22

23 Proposition of design curve for impulse transmission: exponential shock VBP case 23

RPPL (Rigid perfectly plastic locking) Model Face-sheets are assumed to be rigid.

24 Extension of GM Theory for shocks to sandwich composite panels Necessity of core compression model Core compression reduces back facesheet velocity. Advantage due tofsi at the backis overestimated. RPPL (Rigid perfectly plastic locking) Model Face-sheets are assumed to be rigid. Elastic deformation of the core is neglected. Core becomes rigid after densification. Refer: Ghoshal and Mitra (2013), Journal of Applied Physics, Accepted 24

25 RPPL model used in studying impact and shock problems 25

26 Core compression model considering coupled effect of FSI at rear side of the plate Water-backed Air-backed Necessary condition for plastic shock initiation within core Equation of motion Jump condition Assumption: Shock is arrested within the core

27 Derivation of Equation of motion and Jump conditions Conservation of linear momentum, Lagrangian/material description, Small deformation Integration over partial domains Kinematic compatibility condition for discontinuity - Hadamard Lagrangian/material description, Small deformation separated by the plastic shock front discontinuity Rate of change of linear momentum yields equation of motion

28 Results

29 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

30 Results

Simplified model for predicting impulsive loads on submerged structures to account for fluid-structure interaction

Simplified model for predicting impulsive loads on submerged structures to account for fluid-structure interaction Timon Rabczuk, Esteban Samaniego, Ted Belytschko Department of Mechanical Engineering,