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Research Articles

Analytical Modeling of the Underwater Shock Response of Rigid and Elastic Plates Near a Solid Boundary

[+] Author and Article Information
Qinyuan Li

University of Michigan,
Ann Arbor, MI 48109;
Norwegian University of Science and Technology,
7034 Trondheim, Norway
e-mail: qinyuan.li@ntnu.no

Michail Manolidis

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: mihalis@umich.edu

Yin L. Young

Department of Naval Architecture and
Marine Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ylyoung@umich.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received October 3, 2011; final manuscript received August 31, 2012; accepted manuscript posted September 10, 2012; published online January 25, 2013. Assoc. Editor: Vikram Deshpande.

J. Appl. Mech 80(2), 021017 (Jan 25, 2013) (8 pages) Paper No: JAM-11-1352; doi: 10.1115/1.4007586 History: Received October 03, 2011; Revised August 31, 2012; Accepted September 10, 2012

In this paper, analytical solutions are derived for the case when an elastic water-backed plate (WBP) is subject to an exponential shock loading near a fixed solid boundary. Two cases, a rigid plate and an elastic plate represented by two mass elements connected by a spring and a dashpot, are studied. The analytical solution is extended from Taylor's (1963, “The Pressure and Impulse of Submarine Explosion Waves on Plates,” Scientific Papers of Sir Geoffrey Ingram Taylor, Vol. 3, G. K. Batchelor, ed., Cambridge University Press, Cambridge, UK, pp. 287–303) floating air-backed plate (ABP) model and the water-backed plate model of Liu and Young (2008, “Transient Response of Submerged Plates Subject to Underwater Shock Loading: An Analytical Perspective,” J. Appl. Mech., 75(4), 044504; 2010, “Shock-Structure Interaction Considering Pressure Precursor,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, CA). The influences of five parameters are studied: (a) the distance of the fixed boundary from the back plate d, (b) the fluid structure interaction (FSI) parameter φ of the plate, (c) the stiffness of the plate as represented by the natural frequency of the system T, (d) the material damping coefficient CD of the plate, and (e) the pressure precursor (rise) time θr. The results show that the pressure responses at the front and back surfaces of the plate are greatly affected by the proximity to the fixed boundary, the fluid-structure interaction parameter, the ratio of the shock decay time to the natural period of the structure, and the rise time of incident pressure. The effect of structural damping (assuming a typical material damping coefficient of 5%) is found to be practically negligible compared to the other four parameters.

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References

Taylor, G. I., 1963, “The Pressure and Impulse of Submarine Explosion Waves on Plates,” Scientific Papers of Sir Geoffrey Ingram Taylor, Vol. 3, G. K.Batchelor, ed., Cambridge University Press, Cambridge, England, pp. 287–303.
Liu, Z., and Young, Y. L., 2008, “Transient Response of Submerged Plates Subject to Underwater Shock Loading: An Analytical Perspective,” J. Appl. Mech., 75(4), p. 044504. [CrossRef]
Liu, Z., and Young, Y. L., 2010, “Shock-Structure Interaction Considering Pressure Precursor,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, CA, September 12–17.
Zhang, M., and Batra, R. C., 2009, “Analytical and Numerical Solution to the Dynamic Response of Mass Spring Damper System Subject to Underwater Shock Loading,” Proceedings of the 3rd International Conference on Mechanical Engineering and Mechanics, Beijing, China, October 21–23.
Makinen, K., 1998, “Cavitation Models for Structures Excited by a Plane Shock Wave,” J. Fluids Struct., 12(1), pp. 85–101. [CrossRef]
Makinen, K., 1999, “The Transverse Response of Sandwich Panels to an Underwater Shock Wave,” J. Fluids Struct., 13, pp. 631–646. [CrossRef]
Sprague, M. A., and Geers, T. L., 2004, “A Spectral-Element Method for Modelling Cavitation in Transient Fluidastructure Interaction,” Int. J. Numer. Methods Eng., 60, pp. 2467–2499. [CrossRef]
Sprague, M. A., and Geers, T. L., 2006, “A Spectral-Element/Finite-Element Analysis of a Ship-Like Structure Subjected to an Underwater Explosion,” Comput. Methods Appl. Mech. Eng., 195(17–18), pp. 2149–2167. [CrossRef]
Liu, Z., Xie, W. F., and Young, Y. L., 2009, “Numerical Modeling of Complex Interactions Between Underwater Shocks and Composite Structures,” Comput. Mech., 43(2), pp. 239–251. [CrossRef]
Deshpande, V. S., Heaver, A., and Fleck, N. A., 2006, “An Underwater Shock Simulator,” Proc. R. Soc. London, Ser. A, 462, pp. 1021–1041. [CrossRef]
Young, Y. L., Liu, Z., and Xie, W. F., 2009, “Fluid-Structure and Shock-Bubble Interaction Effects During Underwater Explosions Near Composite Structures,” J. Appl. Mech., 76(5), p. 051303. [CrossRef]
Espinosa, H. D., Lee, S., and Moldovan, N., 2006, “A Novel Fluid Structure Interaction Experiment to Investigate Deformation of Structural Elements Subjected to Impulsive Loading,” Exp. Mech., 46(6), pp. 805–824. [CrossRef]

Figures

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Fig. 1

Schematic drawing for a water-backed plate near a fixed boundary subject to an exponentially-decaying shock wave from the left hand side

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Fig. 2

Front, back, net pressure, and velocity histories of a rigid plate with infinite distance from the fixed boundary (φ=1.75, d=∞)

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Fig. 3

Front, back, net pressure, and velocity histories of a rigid plate with distance of d=cθ from the fixed boundary (d=cθ, φ=1.75)

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Fig. 4

Front and back pressure histories of a rigid plate with different distance from the fixed boundary (φ=1.75)

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Fig. 5

Velocity history of a rigid plate with different distance from the fixed boundary (φ=1.75)

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Fig. 6

Front and back pressure histories of a rigid plate near a fixed boundary with different FSI values (d=cθ)

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Fig. 7

Front and back pressure histories of elastic plates near a fixed boundary with different ratio of natural period to pressure decay time. The dotted line is the result of rigid plate model (d=cθ, φ1=φ2=2φ=3.5)

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Fig. 8

Front and back velocity histories of elastic plates near a fixed boundary with different ratio of natural period to pressure decay time. The dotted line is the result of rigid plate model (d=cθ, φ1=φ2=2φ=3.5).

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Fig. 9

Front and back pressure histories of a rigid plate subject to incident pressure with and without precursor. The dotted line is the result without rigid boundary. (d=cθ, φ=1.75, θr/θ=0.1).

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Fig. 10

Net pressure histories of a rigid plate subject to incident pressure with and without precursor. The dotted line is the result without rigid boundary (d=cθ, φ=1.75, θr/θ=0.1).

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Fig. 11

Displacement and velocity histories of a rigid plate subject to incident pressure with and without precursor. The dotted line is the result without rigid boundary (d=cθ, φ=1.75, θr/θ=0.1).

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