Shock and Vibration

Volume 2015 (2015), Article ID 451583, 11 pages

http://dx.doi.org/10.1155/2015/451583

## An Improved Shock Factor to Evaluate the Shock Environment of Small-Sized Structures Subjected to Underwater Explosion

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China

Received 9 October 2014; Revised 30 March 2015; Accepted 31 March 2015

Academic Editor: Wen Long Li

Copyright © 2015 Wenzheng Zhang and Weikang Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Shock factor is conventionally used to assess the effect of an underwater explosion on a target. The dimensions of some structures are much smaller than the wavelength of incident wave induced by the underwater explosion. The conventional shock factor may be excessively severe for small-sized structures because it neglects the effect of scattering; so it is necessary to study the shock factor for small objects. The coupled mode method is applied to study the scattering field surrounding the cylindrical shells. A nonlinear relation differential is derived from the impact received by the cylindrical shells and the ratio between the diameters of the shells and the wavelength of the incident wave. An improved shock factor is developed based on the fitted curve, considering the scattering effect caused by the diameters of the submerged cylindrical shells. A set of numerical simulations are carried out to validate the accuracy of the proposed approach. The results show that the cylindrical shells and spherical shells under different conditions, but with the same shock factor, have almost the same shock responses.

#### 1. Introduction

The dynamic responses and shock damage of the vessels subjected to underwater explosion (UNDEX) are important concerns for naval researchers. Cylindrical shells and spherical shells account for a large proportion in the vessels and offshore structures, especially in surface ships and submarines. Therefore, the researches on the dynamic responses and shock environment of the cylindrical shells subjected to underwater shock load make great significance in assessing the impact resistance ability of submerged structures.

Full scale test is the most direct and effective method to study how the vessels dynamically respond to and are damaged by the underwater shock wave. However, conducting experiments on full scale vessels is extremely expensive and the budget is generally limited. Scaled models are good alternatives for cost reduction and convenient manipulation. Cole [1] systematically summarized the theoretical achievements and test phenomena related to UNDEX, and the empirical formulas to predict the shock wave loading and dynamic responses of the structures were also simplified. Brett et al. [2] investigated the dynamic responses of the steel cylinders exposed to near-field explosion by measuring the acceleration and underwater pressure. Li et al. [3] compared the linear and nonlinear responses and damage modes of the unfilled and main hull sand-filled cylindrical shell models subjected to underwater spherical explosion by a series of small-scale experiments. Chen et al. [4] experimentally investigated the strain and acceleration responses of a neoprene coated cylinder subjected to UNDEX in an artificial lake.

Developing an analytical solution to the UNDEX problem is extremely difficult because the dynamic responses of the vessels depend on many factors, including the complex structures, the detonation of the explosive charge, the propagation of the shock wave, the local cavitation, and the complicated fluid-structure interaction (FSI). Therefore, the theoretical studies are only suitable for simple geometrical structures. Huang [5, 6] employed the series expansion method and Laplace transform to elucidate the transient FSI of plane acoustic waves with submerged spherical and cylindrical elastic shells. Geers [7, 8] obtained the responses of the submerged cylindrical shell exited by a transient plane wave by using the residual potential method. Zhang and Geers [9] presented the transient response histories of the submerged fluid-filled spherical shell exposed to a plane step wave. The separation of variables method was adopted and the results were compared with that of an empty submerged steel shell.

Over the last decades, with the improvement of computer technique and calculation procedure, a variety of numerical methods have been rapidly developed to analyze the FSI problem. The development of the finite element method (FEM) and boundary element method (BEM) makes it possible to investigate the responses of submerged complex structures subjected to noncontact UNDEX, and the small-scale model tests are frequently carried out to validate the feasibility of the numerical methods. Kwon and Fox [10] studied the nonlinear dynamic response of a cylinder subjected to a side-on, far-field UNDEX by using both numerical and experimental techniques. Wardlaw and Luton [11] presented several close-in cases to document the FSI mechanisms for internal and external UNDEX, and the rigid and deformable body simulations were compared by applying the coupled GEMINI–DYNA_N code. Mair [12] indicated that only codes employing “structural elements” were realistically applicable to the analysis of thin-walled structural response to UNDEX after reviewing the applicability of various hydrocode methodologies. Hung et al. [13] analyzed the linear and nonlinear dynamic responses of three cylindrical shell structures (unstiffened, internal, and external stiffened) subjected to small charge UNDEX in a water tank, and the dynamic accelerations and strains were compared with those obtained by FEM. Jin and Ding [14] compared the dynamic acceleration and velocity responses of a ship section between the experimental and numerical results by using ABAQUS.

Among the numerical techniques developed for the FSI problem, particularly worth mentioning is the doubly asymptotic approximation (DAA) method. Geers [15, 16] applied approximations approach in the limit of low and high frequency motions by using the virtual mass and plane wave approximation, and a smooth transition was effected in the intermediate frequency range to analyze the dynamic responses of the submerged structures subjected to UNDEX. Geers and Felippa [17] also studied the accuracy of the DAA forms through the numerical results of a submerged spherical shell. Liang and Tai [18] presented time history of the shock wave and the dynamic responses of a patrol boat subjected to underwater shock loading by using the FEM coupled with DAA. Lai [19] applied the time domain FEM/DAA coupling procedure to predict the transient dynamic responses of a submerged sphere shell with an opening subjected to UNDEX, and the results in the sea and air were also compared.

Most of the previous research mainly focused on the deformation, damage, and buckling of the cylindrical shells subjected to the shock wave. However, it is difficult to analyze the scattered wave of structures by the fluid-structure interaction method, and there are few studies on the incident wave in analytical method. Therefore, the research on the scattered wave field of submerged structures by the analytical methods seems to be very imperative. When the characteristic dimensions of small-sized submerged structures, such as towed vehicles and torpedoes, are much smaller than the wavelength, the scattering effect should be taken into consideration. In this paper, the coupled mode method is applied to decompose the incident wave into a series of harmonic waves. The scattering effect of submerged cylindrical shells is analyzed to verify that the conventional shock factor is excessively severe for small-sized structures. Therefore, the conventional shock factor is revised according to the impulse received by the cylindrical shell, which takes in the scattering effect of the characteristic dimensions of submerged structures. A set of numerical simulations are carried out by using the commercial code to validate the accuracy of the improved shock factor. Results show that the responses of cylindrical shells and spherical shells in different conditions are similar to each other when the improved shock factor is unchanged.

#### 2. Shock Wave Pressure of UNDEX

UNDEX is the major threat to surface ships and submarines. According to the dynamic responses and damage modes of the vessels, the noncontact UNDEX can be divided into two kinds: near-field explosion and far-field explosion. In a near-field UNDEX, the structures are within the maximum radius of the first pulsation of the gas bubble, and the shock energy of the explosive charge may cause great local damage to the vessels. For far-field UNDEX, the standoff distance between the explosive charge and the structures is larger than the maximum radius , and the explosion can cause a wide range of nonrepair damage and failure of shipboard equipment. So the majority of the previous studies focus on the far-field explosion.

During an UNDEX, the sudden release of the explosive energy generates a transitory and highly compressed shock wave and a series of gas bubble pulsations. Most tests indicate that the damage and failure of the vessels occur at the early time of an UDNEX and are caused by the primary shock wave. The energy of the shock wave delivered to the vessels depends on the explosive charge weight and standoff distance. The shock wave is superimposed onto the hydrostatic pressure and propagates into the water medium in a spherical shape. The time history of the shock wave at a fixed location starts with an instantaneous peak pressure in time domain, followed by an exponentially decaying function. According to the empirical formula summarized by Cole [1] and Zamyshlyaev [20], the incident shock wave can be expressed bywhere denotes the peak magnitude of the pressure at the shock front; represents the time decay constant and is the propagation time from the explosive to the target. For trinitrotoluene (TNT), the peak pressure and the time decay constant arewhere is the explosive charge weight, is the standoff distance, and , , , and are the shock parameters of the explosive.

#### 3. Coupled Mode Method

Cylindrical shells account for a large proportion of submerged structures. Many analytical approaches to the scattered wave field of cylindrical shells have been developed, such as the coupled mode method, the finite difference time domain method, and the reflected afterflow of virtual source method. In this paper, the coupled mode method is introduced to compute the scattered wave field by three-dimensional cylindrical shells for reducing computation time and ensuring precision.

A variety of researches on scattered wave field of harmonic wave have been conducted [21], but the incident wave induced by UNDEX is not a harmonic wave. It is necessary to decompose the incident wave into a series of harmonic plane waves with various frequencies. The incident wave can be expressed as

Assume ; then (3) is transformed into

The Fourier transform of (4) can be obtained:

The inverse Fourier transform of is yielded as

The incident wave can be regarded as plane wave, as the standoff distance is usually very large for the small-sized submerged structures. Hence, substituting (4) into (5) yields

Substituting (7) into (6) yieldswhere .

The schematic diagram of scattered wave field of submerged cylindrical shell is shown in Figure 1, the explosive charge is located at point , and the incident wave propagates along the -axis direction. For conveniently computing the scattered wave field of cylindrical shell, the Cartesian coordinates are transformed into cylindrical coordinates .