An Analytical Method for Evaluating the Dynamic Response of Plates Subjected to Underwater Shock Employing Mindlin Plate Theory and Laplace Transforms

Abstract

It is often in the interest of a designer to know the transient state of stress in a plate subjected to an underwater explosion. In this paper, an analytical method based on Taylor’s fluid-solid interaction (FSI) model, Mindlin plate theory, Laplace transform, and its inversion is proposed to examine the elastic dynamic response of a plate subjected to an underwater explosion. This analytical method includes shear deformation, the moments and membrane stress in the plate, and the FSI effect and considers a full profile of possibilities. The results of the response-time histories and the response distribution on the plate in terms of displacements and stresses from the analytical method are compared with finite element analysis (FEA) to validate this method, and the comparison indicates good agreement. Comparison of the acceleration at the center of an air-backed plate between the analytical method and the experiment from relevant literature, shows good agreements, and the analytical method and its FSI model are validated. The influence of the FSI is investigated in detail. All extreme values of the response-time histories decrease as the thickness increases for the non-FSI case. The results can be used as benchmark solutions in further research.

1. Introduction

Plate element is one of the basic elements of every class of ship and apparatus and may be subjected to underwater explosions. It is in the interest of a ship designer to know the transient state of stress in advance that develops in a plate during an underwater explosion. For low-intensity explosions, the stresses developed within a plate are usually within the material’s elastic range [1–3].

The classical plate theory (CPT) has been used to solve the response in thin plates to an underwater explosion. An orthotropic and simply supported plate reinforced by stiff particles embedded in a matrix offers the potential for simple, economically functional grading and enhanced response. Genin and Birman [4] obtained a solution for the response of this plate under a uniform and time-varying overpressure using the Kirchhoff plate theory and a convolution integral. However, the shear deformation through the thickness is neglected within CPT, and the effect of fluid-solid interaction (FSI) was not considered in this research.

Some of recent research on this subject have been devoted to the extension of linear to nonlinear response by involving von Kármán thin plate theory and employing the extended Galerkin and the Runge-Kutta methods. Hause [5] investigated the nonlinear response of a functionally graded plate with two constituent phases exposed to a Friedlander explosive air blast within the classical plate theory, but the effect of fluid-solid interaction (FSI) was not considered in this research. Librescu et al. [6–9] have developed a three-dimensional sandwich model which is used in the investigation of the dynamic responses of a sandwich plate subjected to underwater explosions by employing Hayman’s FSI model [6, 7]. In the sandwich model, core layer can only carry transverse shear stresses, while the transverse shear effects are neglected for the face sheets. In general, all of above research concerning the nonlinear response are numerical but not analytical.

The phenomenon of FSI significantly influences the dynamic response of a structure during an underwater explosion. The influence of FSI is typically investigated by finite element analysis (FEA) which provides a detailed picture depending on the level of complexity of the model [10–13]. The discretization used both for spatial derivation and time integration, however, affects spurious oscillations and the accuracy of a numerical solution [14].

Taylor [15] proposed a one-dimensional FSI model for a planar shock wave impinging on a freestanding plate and considered the momentum transmitted to the plate by the shock wave. Liu and Young [16] extended Taylor’s air-backed FSI model to a water-backed FSI model and systematically investigated the influence of the back conditions on the interaction between the fluid and the structure. Utilizing Taylor’s FSI model, the energy method, and considering the conditions of the air-backed and water-backed cases, Rajendran et al. [1–3] derived one-dimensional, semianalytical models to predict the elastic strains in circular and rectangular plates subjected to underwater explosions. Good agreement was found between the model and the experimental results. However, these solutions considering the FSI effect are one-dimensional and ignore moment and membrane stress in the plate. In addition to elastic method, plastic dynamic analytical solutions for the response of a three-layered sandwich plate with a soft core to UNDEX have also formed the subject of recent studies [11, 17–20], in which the existing plastic solution procedure generally comprises three stages. However, the velocity is assumed to be uniform for the front and back faces of the sandwich plate except for the boundaries in the FSI phase, with the FSI differences between the different positions on front and back faces of plate being ignored, that is to say, only one-dimensional FSI effect is considered.

A comprehensive review on the methods for the response of plates to underwater explosion is present above. It is worthy noted that the analytical methods mentioned above do not consider the FSI differences between the different positions on front and back faces of plate. The purpose of this paper is to fill this gap by proposing an analytical method to determine the elastic dynamic response of a plate subjected to an underwater explosion based on Taylor’s FSI model, Mindlin plate theory, Laplace transform, and its inversion. This method considers the full profile of possibilities, such as non-FSI, air-backed, and water-backed cases. Section 1 introduces the solution methods. In Section 2, the problem is described and Taylor’s FSI model, the air-backed, and the water-backed cases are introduced. Section 3 introduces the governing equations of motion for Mindlin plate theory. In Section 4, analytical solutions for the three cases are derived by Fourier transform, Laplace transform, and their inversion. In Section 5, the analytical results are validated by being compared with those from the FEA and the experiment record, and the influences of the FSI and material thickness are investigated. The conclusions are discussed in Section 6.

2. Problem Description and Fluid-Solid Interaction

Consider a rectangular plate of length 𝑎, width 𝑏, and a uniform thickness ℎ, as shown in Figure 1. The plate is subjected to an underwater explosion from an explosive charge 𝑊 (TNT equivalent weight in kg) located at a distance 𝑆 (in 𝑚) from the center of the plate.

Figure 1 

A rectangular plate subjected to an underwater explosion showing dimensions and the Cartesian coordinate system.

Taylor [15] proposed a one-dimensional FSI model for a planar shock wave impinging on a free-standing plate, including the momentum transmitted to the plate by the shock wave. This FSI model has been widely used in analytical methods modeling the plastic response of plates subjected to an underwater explosion [17, 19, 20] and is briefly reviewed below.

Adopting the planar wave assumption, the water pressure 𝑝_𝑓(𝑥,𝑦,𝑡) at the front face (𝑧=−ℎ/2) comprises the incident wave 𝑝_𝑖(𝑥,𝑦,𝑡), the idealized perfectly reflected wave 𝑝_𝑟1(𝑥,𝑦,𝑡), and the rarefaction wave pressure 𝑝_𝑟2(𝑥,𝑦,𝑡), respectively, as follows [15]: 𝑝_𝑓𝑥,𝑦,𝑡=𝑝_𝑖𝑥,𝑦,𝑡+𝑝_𝑟1𝑥,𝑦,𝑡+𝑝_𝑟2𝑥,𝑦,𝑡=2𝑝_𝑚−𝑡exp𝑡_𝑑−𝜌_𝑤𝑐_𝑤𝑤̇ℎ𝑥,𝑦,𝑧=−2,𝑡, (1) where 𝑤 denotes the deflection of the plate and the dots over a quantity stand for time derivatives.

The peak pressure 𝑝_𝑚 and the time parameter 𝑡_𝑑 of the incident wave are given as follows [1, 21]: 𝑝_𝑚𝑊=52.16×^1/3𝑆^1.13𝑡MPa,_𝑑=96.5×10⁻³×𝑊^1/3×𝑊^1/3𝑆^−0.22msec. (2)

Two backing condition categories are water-backed and air-backed. For an air-backed plate, the water pressure at the back face (𝑧=ℎ/2) is [1] 𝑝_𝑏𝑥,𝑦,𝑡=0, (3) and for a water-backed plate, the pressure at the back face is [1] 𝑝_𝑏𝑥,𝑦,𝑡=−𝜌_𝑤𝑐_𝑤𝑤̇ℎ𝑥,𝑦,𝑧=2,𝑡. (4)

To compare the two cases with the FSI effects between the plate and the water, a non-FSI case (ignoring FSI effects) is also discussed. The water pressures at both front and back faces can be expressed as follows [1]: 𝑝_𝑓𝑥,𝑦,𝑡=𝑝_𝑚−𝑡exp𝑡_𝑑,𝑝_𝑏𝑥,𝑦,𝑡=0. (5)

3. Fundamental Equations

Considering an orthotropic material, the Mindlin plate theory implies that the displacements are distributed in the plate as follows: 𝑢=−𝑧𝜓_𝑥𝑧,𝑣=−𝑧𝜓_𝑦𝑧,𝑤=𝑤₀, (6) where 𝑤₀ is the vertical displacement at the middle surface.

The moment and shear force resultants can be written in terms of the displacements as 𝑀_𝑥𝑀_𝑦𝑀_𝑥𝑦=ℎ³𝑐12¹¹𝑐₁₂0𝑐₁₂𝑐₂₂000𝑐₆₆−𝜓_{𝑥𝑧,𝑥}−𝜓_{𝑦𝑧,𝑦}−𝜓_{𝑥𝑧,𝑦}+𝜓_{𝑦𝑧,𝑥}2,𝑄_𝑦𝑄_𝑥=𝜅ℎ2𝑐₄₄00𝑐₅₅𝑤_0,𝑦−𝜓_𝑦𝑧𝑤_0,𝑥−𝜓_𝑥𝑧, (7) where 𝜓_𝑥𝑧 and 𝜓_𝑦𝑧 are the rotations about the 𝑥- and 𝑦-axes, respectively, ℎ is the thickness of the plate, 𝜅 is the shear correction factor (taken as 5/6), and 𝑐₁₁, 𝑐₁₂, 𝑐₂₂, 𝑐₄₄, 𝑐₅₅, and 𝑐₆₆ are the elastic constants. Throughout the paper, a comma after a quantity denotes a particular differentiation with respect to the spatial coordinates.

The normal stresses are distributed as 𝜎_𝑥𝑥=12𝑧𝑀_𝑥ℎ³,𝜎_𝑦𝑦=12𝑧𝑀_𝑦ℎ³,𝜎_𝑧𝑧=0, (8) and the shear stresses are distributed as 𝜎_𝑥𝑦=12𝑧𝑀_𝑥𝑦ℎ³,𝜎_𝑥𝑧=31−4𝑧²/ℎ²𝑄_𝑥(),𝜎2ℎ_𝑦𝑧=31−4𝑧²/ℎ²𝑄_𝑦().2ℎ (9)

The von Mises stress 𝜎_𝑚 can be expressed as 𝜎_𝑚=𝜎_𝑥𝑥−𝜎_𝑦𝑦²+𝜎_𝑥𝑥−𝜎_𝑧𝑧²+𝜎_𝑦𝑦−𝜎_𝑧𝑧²+6𝜎²_𝑥𝑦+𝜎²_𝑥𝑧+𝜎²_𝑦𝑧2. (10)

The governing equations of motion for the Mindlin plate theory are 𝑀_𝑥,𝑥+𝑀_{𝑥𝑦,𝑦}+𝜌_𝑝𝐽𝜓̈_𝑥𝑧=𝑄_𝑥,𝑀_{𝑥𝑦,𝑥}+𝑀_𝑦,𝑦+𝜌_𝑝𝐽𝜓̈_𝑦𝑧=𝑄_𝑦,𝑄_𝑥,𝑥+𝑄_𝑦,𝑦−𝜌_𝑝ℎ𝑤̈₀+𝑝_𝑓+𝑝_𝑏.=0, (11) where 𝐽=ℎ³/12.

Substituting (7) into (11) allows (11) to be rewritten as ℎ³2𝑐₁₁𝜓_{𝑥𝑧,𝑥𝑥}+𝑐₁₂𝜓_{𝑦𝑧,𝑥𝑦}+𝑐₆₆𝜓_{𝑥𝑧,𝑦𝑦}+𝜓_{𝑦𝑧,𝑥𝑦}−ℎ24₃𝜌_𝑝𝜓̈_𝑥𝑧+𝑐12⁵⁵𝜅ℎ𝑤_0,𝑥−𝜓_𝑥𝑧2ℎ=0,₃2𝑐₁₂𝜓_{𝑥𝑧,𝑥𝑦}+𝑐₂₂𝜓_{𝑦𝑧,𝑦𝑦}+𝑐₆₆𝜓_{𝑥𝑧,𝑥𝑦}+𝜓_{𝑦𝑧,𝑥𝑥}−ℎ24₃𝜌_𝑝𝜓̈_𝑦𝑧+𝑐12⁴⁴𝜅ℎ𝑤_0,𝑦−𝜓_𝑦𝑧2𝜌=0,_𝑝ℎ𝑤̈₀−𝜅ℎ𝑐₅₅𝑤_0,𝑥𝑥−𝜓_{𝑥𝑧,𝑥}+𝑐₄₄𝑤_0,𝑦𝑦−𝜓_{𝑦𝑧,𝑦}2=𝑝_𝑓+𝑝_𝑏. (12)(13)(14)

Assuming the plate is simply supported, the edge conditions are given as 𝑥=0,𝑎∶𝑤₀=𝜓_𝑦𝑧=𝑀_𝑥=0,𝑦=0,𝑏∶𝑤₀=𝜓_𝑥𝑧=𝑀_𝑦=0. (15)

The following Fourier series expansions for the displacements satisfy (15): 𝑤₀=^∞∞_𝑚=1𝑛=1𝑤₀sin𝛼_𝑚𝑥sin𝛽_𝑛𝜓𝑦,_𝑥𝑧=^∞∞_𝑚=1𝑛=1𝜓_𝑥𝑧cos𝛼_𝑚𝑥sin𝛽_𝑛𝜓𝑦,_𝑦𝑧=^∞∞_𝑚=1𝑛=1𝜓_𝑦𝑧sin𝛼_𝑚𝑥cos𝛽_𝑛𝑦, (16) where 𝛼_𝑚=𝑚𝜋𝑎,𝛽_𝑛=𝑛𝜋𝑎, (17) and 𝑚,𝑛=1,2… are the number of half waves in the 𝑥 and 𝑦 directions, respectively, and the overscript (^--) denotes the transformed function in the Fourier domain.

According to the plane wave assumption, the incident wave and the idealized perfectly reflected wave can also be expanded in Fourier series form as 𝑝_𝑖=^∞∞_𝑚=1𝑛=1𝑝_𝑖sin𝛼_𝑚𝑥sin𝛽_𝑛𝑦,𝑝_𝑖=4𝑎𝑏^𝑎₀^𝑏₀𝑝_𝑖sin𝛼_𝑚𝑥sin𝛽_𝑛𝑦𝑑𝑥𝑑𝑦=4𝜂₁𝑝_𝑖,𝑝𝑎𝑏_𝑟1=^∞∞_𝑚=1𝑛=1𝑝_𝑟1sin𝛼_𝑚𝑥sin𝛽_𝑛𝑦,𝑝_𝑟1=4𝑎𝑏^𝑎₀^𝑏₀𝑝_𝑟1sin𝛼_𝑚𝑥sin𝛽_𝑛𝑦𝑑𝑥𝑑𝑦=4𝜂₁𝑝_𝑟1,𝑎𝑏 (18)(19)(20)(21) where 𝜂₁=−1+(−1)^𝑚−1+(−1)^𝑛𝑚𝑛𝜋². (22)

4. Solutions Procedure

The analytical solutions in the Fourier and Laplace transformed domains are derived for the three cases, respectively. The final solutions to the displacement distributions are obtained by the inversion of the Fourier and Laplace transforms. The inversion of Laplace transform is carried out analytically.

4.1. Non-FSI

For the non-FSI case, by substituting (5) and (16)–(19) into (12)–(14), the fundamental equation can be rewritten in the Fourier transformed domain as follows: ℎ³𝜌_𝑝𝜓̈‌_𝑥𝑧+12𝜓_𝑥𝑧ℎ³2𝑐₁₁𝛼²_𝑚+𝑐₆₆𝛽²_𝑛+𝑐24⁵⁵ℎ𝜅2+2𝑐₁₂+𝑐₆₆ℎ³𝛼_𝑚𝛽_𝑛𝜓_𝑦𝑧−𝑐24⁵⁵ℎ𝜅𝑤₀𝛼_𝑚2ℎ=0,₃𝜌_𝑝𝜓̈‌_𝑦𝑧+12𝜓_𝑦𝑧ℎ³𝑐₆₆𝛼²_𝑚+2𝑐₂₂𝛽²_𝑛+𝑐24⁴⁴ℎ𝜅2+2𝑐₁₂+𝑐₆₆ℎ³𝛼_𝑚𝛽_𝑛𝜓_𝑥𝑧−𝑐24⁴⁴ℎ𝜅𝑤₀𝛽_𝑛2=0,ℎ𝜌_𝑝𝑤̈‌₀+ℎ𝜅𝑤₀𝑐₅₅𝛼²_𝑚+𝑐₄₄𝛽²_𝑛2−𝑐₅₅ℎ𝜅𝛼_𝑚𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2=4𝜂₁𝑝_𝑚exp−𝑡/𝑡_𝑑.𝑎𝑏 (23)(24)(25)

The Laplace transform of the function 𝑓(𝑡) is defined as 𝐿𝑓(𝑡)=𝑓(𝑠)=^∞₀𝑓(𝑡)𝑒^−𝑠𝑡𝑑𝑡, (26) where 𝑠 is the parameter of the Laplace transform and the overscript (^~) denotes the transformed function in the Laplace domain. Applying the Laplace transform to (23)–(25) yields: ℎ³2𝑐₁₁𝛼²_𝑚+𝑐₆₆𝛽²_𝑛+2𝑠²𝜌_𝑝+𝑐24⁵⁵ℎ𝜅2𝜓_𝑥𝑧+ℎ³𝛼_𝑚𝛽_𝑛2𝑐₁₂+𝑐₆₆𝜓_𝑦𝑧−𝑐24⁵⁵ℎ𝜅𝛼_𝑚𝑤₀2ℎ=0,₃𝑐₆₆𝛼²_𝑚+2𝑐₂₂𝛽²_𝑛+2𝑠²𝜌_𝑝+𝑐24⁴⁴ℎ𝜅2𝜓_𝑦𝑧+ℎ³𝛼_𝑚𝛽_𝑛2𝑐₁₂+𝑐₆₆𝜓_𝑥𝑧−𝑐24⁴⁴ℎ𝜅𝛽_𝑛𝑤₀2ℎ=0,𝑤₀𝑐₅₅𝜅𝛼²_𝑚+𝑐₄₄𝜅𝛽²_𝑛+2𝑠²𝜌_𝑝2−𝑐₅₅ℎ𝜅𝛼_𝑚𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2=4𝜂₁𝑡_𝑑𝑝_𝑚𝑎𝑏1+𝑠𝑡_𝑑. (27)(28)(29)

Solutions in the Fourier and Laplace transformed domains for the non-FSI case can be derived from (27)–(29) as follows: 𝑤₀=16𝑝_𝑚𝑡_𝑑𝜂₁𝜂²₃−𝜂₂𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₆,𝜓_𝑥𝑧=8ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₄₄𝛽_𝑛𝜂₃−𝑐₅₅𝛼_𝑚𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₆,𝜓_𝑦𝑧=8ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₅₅𝛼_𝑚𝜂₃−𝑐₄₄𝛽_𝑛𝜂₂𝑎𝑏1+𝑠𝑡_𝑑𝜂₆, (30) where 𝜂₂=ℎ𝜅𝑐₅₅2+ℎ³2𝑐₁₁𝛼²_𝑚+𝑐₆₆𝛽²_𝑛+2𝑠²𝜌_𝑝,𝜂24₃=ℎ³2𝑐₁₂+𝑐₆₆𝛼_𝑚𝛽_𝑛,𝜂24₄=ℎ𝜅𝑐₄₄2+ℎ³𝑐₆₆𝛼²_𝑚+2𝑐₂₂𝛽²_𝑛+2𝑠²𝜌_𝑝,𝜂24₅=ℎ𝜅𝑐₅₅𝛼²_𝑚+𝜅𝑐₄₄𝛽²_𝑛+2𝑠²𝜌_𝑝2,𝜂₆=ℎ²𝜅²𝑐₄₄𝛽_𝑛𝑐₄₄𝛽_𝑛𝜂₂−2𝑐₅₅𝛼_𝑚𝜂₃+𝑐²₅₅𝛼²_𝑚𝜂₄+4𝜂²₃−𝜂₂𝜂₄𝜂₅. (31)

4.2. Air-Backed

Considering the FSI effects for an air-backed plate case and substituting (1), (3), and (16)–(19) into (12)–(14), the fundamental (12)-(13) that are rewritten in the Fourier transformed domain are the same as (23)-(24). These rewritten equations in both the Fourier and Laplace transformed domains are the same as (27)-(28), respectively. Moreover, (14) can be rewritten as follows: 𝑐_𝑤𝜌_𝑤𝑤̇‌₀+ℎ𝜌_𝑝𝑤̈‌₀−𝑐₅₅ℎ𝜅𝛼_𝑚𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2+ℎ𝜅𝑤₀𝑐₅₅𝛼²_𝑚+𝑐₄₄𝛽²_𝑛2=8𝜂₁𝑝_𝑚exp−𝑡/𝑡_𝑑.𝑎𝑏 (32)

Applying the Laplace transform to (32) yields: 𝑤₀ℎ𝑐₅₅𝜅𝛼²_𝑚+𝑐₄₄𝜅𝛽²_𝑛+2𝑠²𝜌_𝑝2+𝑠𝑐_𝑤𝜌_𝑤−𝑐₅₅ℎ𝜅𝛼_𝑚×𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2=8𝜂₁𝑡_𝑑𝑝_𝑚𝑎𝑏𝑠𝑡_𝑑.+1 (33)

Solving (27), (28), and (33) yields the analytical solutions in the Fourier and Laplace transformed domains for an air-backed plate: 𝑤₀=32𝑝_𝑚𝑡_𝑑𝜂₁𝜂²₃−𝜂₂𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₈,𝜓_𝑥𝑧=16ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₄₄𝛽_𝑛𝜂₃−𝑐₅₅𝛼_𝑚𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₈,𝜓_𝑦𝑧=16ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₅₅𝛼_𝑚𝜂₃−𝑐₄₄𝛽_𝑛𝜂₂𝑎𝑏1+𝑠𝑡_𝑑𝜂₈, (34) where 𝜂₇=ℎ𝜅𝑐₅₅𝛼²_𝑚+𝜅𝑐₄₄𝛽²_𝑛+2𝑠²𝜌_𝑝2+𝑠𝑐_𝑤𝜌_𝑤,𝜂₈=ℎ²𝜅²𝑐₄₄𝛽_𝑛𝑐₄₄𝛽_𝑛𝜂₂−2𝑐₅₅𝛼_𝑚𝜂₃+𝑐²₅₅𝛼²_𝑚𝜂₄+4𝜂²₃−𝜂₂𝜂₄𝜂₇. (35)

4.3. Water-Backed

Similarly to the air-backed case, for the water-backed plate case, the fundamental (12)-(13) are rewritten in both Fourier and Laplace transformed domains and are the same as (27)-(28). Moreover, by substituting (1), (4), and (16)–(19) into (14), this equation can be rewritten as 2𝑐_𝑤𝜌_𝑤𝑤̇‌₀+ℎ𝜌_𝑝𝑤̈‌₀−𝑐₅₅ℎ𝜅𝛼_𝑚𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2+ℎ𝜅𝑤₀𝑐₅₅𝛼²_𝑚+𝑐₄₄𝛽²_𝑛2=8𝜂₁𝑝_𝑚−𝑡exp𝑡_𝑑.𝑎𝑏 (36)

Applying the Laplace transform to (36) yields: 𝑤₀ℎ𝑐₅₅𝜅𝛼²_𝑚+𝑐₄₄𝜅𝛽²_𝑛+2𝑠²𝜌_𝑝2+2𝑠𝑐_𝑤𝜌_𝑤−𝑐₅₅ℎ𝜅𝛼_𝑚×𝜓_𝑥𝑧2−𝑐₄₄ℎ𝜅𝛽_𝑛𝜓_𝑦𝑧2=8𝜂₁𝑡_𝑑𝑝_𝑚𝑎𝑏𝑠𝑡_𝑑.+1 (37)

Solving (27), (28), and (37) yields the analytical solutions in the transformed domains: 𝑤₀=32𝑝_𝑚𝑡_𝑑𝜂₁𝜂²₃−𝜂₂𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₁₀,𝜓_𝑥𝑧=16ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₄₄𝛽_𝑛𝜂₃−𝑐₅₅𝛼_𝑚𝜂₄𝑎𝑏1+𝑠𝑡_𝑑𝜂₁₀,𝜓_𝑦𝑧=16ℎ𝜅𝑝_𝑚𝑡_𝑑𝜂₁𝑐₅₅𝛼_𝑚𝜂₃−𝑐₄₄𝛽_𝑛𝜂₂𝑎𝑏1+𝑠𝑡_𝑑𝜂₁₀, (38) where 𝜂₉=ℎ𝜅𝑐₅₅𝛼²_𝑚+𝜅𝑐₄₄𝛽²_𝑛+2𝑠²𝜌_𝑝2+2𝑠𝑐_𝑤𝜌_𝑤,𝜂₁₀=ℎ²𝜅²𝑐₄₄𝛽_𝑛𝑐₄₄𝛽_𝑛𝜂₂−2𝑐₅₅𝛼_𝑚𝜂₃+𝑐²₅₅𝛼²_𝑚𝜂₄+4𝜂²₃−𝜂₂𝜂₄𝜂₉. (39)

At this point, the analytical solutions for the three cases in the transformed domains have been obtained, and subsequently, the final solutions for the displacement distributions are obtained by the inversion of Fourier and Laplace transforms. The Fourier inversion can be carried out by utilizing (16), and the Laplace inversion can be carried out by adopting an analytical inversion method, which is expressed as follows [23–25].

The solutions for the displacements in (30), (34), and (38) can be rewritten in a general and simpler form as 𝑓(𝑠)=𝐴(𝑠)𝐵(𝑠), (40) where 𝑓(𝑠) denotes a displacement solution in the Laplace transformed domain, both functions 𝐴(𝑠) and 𝐵(𝑠) are in form of polynomials, and the degree of 𝐴(𝑠) is less than that of 𝐵(𝑠).

If all the roots of 𝐵(𝑠) are simple, then the original function is [23–25] 𝑓(𝑡)=^𝑛_𝑘=1𝐴𝑠_𝑘𝐵^′𝑠_𝑘exp𝑠_𝑘𝑡. (41) When 𝐵(𝑠) has complex roots, the original function becomes [23–25] 𝑓(𝑡)=^𝑛𝑟_𝑖=1𝐴𝑠_𝑖𝐵^′𝑠_𝑖exp𝑠_𝑖𝑡+2Re^𝑛𝑐_𝑗=1𝐴𝑠_𝑗𝐵^′𝑠_𝑗exp𝑠_𝑗𝑡, (42) where 𝑠_𝑘, 𝑠_𝑖, and 𝑠_𝑗 are the roots, the real roots, and the complex roots with positive imaginary parts of 𝐵(𝑠), respectively, 𝑛, 𝑛_𝑟, and 𝑛_𝑐 are the numbers of 𝑠_𝑘, 𝑠_𝑖, and 𝑠_𝑗, respectively, the superscript (′) stands for the derivative with respect to 𝑠, and Re stands for the real part of the complex number.

5. Results and Discussions

A Mathematica package was developed for the analytical method described herein. First, for an air-backed plate, the results of the analytical method are validated by a comparison with a finite element analysis (FEA) performed using the commercial software ANSYS. Subsequently, the influences of the FSI and the thickness are investigated.

5.1. Validation of the Analytical Method by Comparing with FE Method

For an air-backed plate, the analytical method is validated by comparing the results of this method with those obtained by the FEA. A simply supported plate is shown in Figure 1, and the corresponding FEA model is shown in Figure 2. The plate is simulated by solid45 elements, and Combin14 damping elements are used to the damping term in (1). The elastic properties, explosive characteristics, and geometric parameters are listed in Tables 1 and 2, where 𝑥_len and 𝑦_len are the element lengths in the 𝑥 and the 𝑦 directions, respectively, of the FEA model for the laminated plate. The variable 𝑧_len is the element length in the 𝑧 direction, and 𝑚_max and 𝑛_max are the maximum values of 𝑚 and 𝑛, respectively.

Table 1 

The parameters of the underwater explosion, geometry for the analytical and FEA methods, and the Fourier transform parameters.

Table 2 

The elastic constants of the plate material.

Figure 2 

The model used for the FEA.

In Figure 3, the response-time histories of 𝑤, 𝜎_𝑥𝑥, 𝜎_𝑦𝑦, 𝜎_𝑥𝑦, 𝜎_𝑥𝑧, 𝜎_𝑦𝑧, and 𝜎_𝑚 of the back face (𝑧=ℎ/2) and middle face (ℎ=0) at Na (𝑥=𝑦=0.5 m), Nb (𝑥=𝑦=0.3 m), and Nc (𝑥=𝑦=0.1 m) are calculated by the analytical and the FEA methods for the air-backed case and are compared. In Figure 4, the distributions of the deflection 𝑤 and the von Mises stress 𝜎_𝑚 on the back face of the plate when 𝑡=0.45 ms are calculated by the analytical and the FEA methods for the air-backed case and are compared. The time-histories and the distributions of the responses between the two solution methods correspond closely.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 3 

Comparisons of the response-time histories calculated by both the analytical and the FEA methods for the air-backed cases: (a) the time histories of the deflection 𝑤 of the back face at Na, Nb, and Nc; (b) the time histories of the membrane normal stress 𝜎𝑥𝑥 of the back face at Na, Nb, and Nc; (c) the time histories of the membrane normal stress 𝜎𝑦𝑦 of the back face at Na, Nb, and Nc; (d) the time histories of the membrane shear stress 𝜎𝑥𝑦 of the back face at Na, Nb, and Nc; (e) the time histories of the transverse shear stress 𝜎𝑥𝑧 of the middle face at Na, Nb, and Nc; (f) the time histories of transverse shear stress 𝜎𝑦𝑧 of the middle face at Na, Nb, and Nc; and (g) the time histories of the von Mises stress 𝜎𝑚 of the back face at Na, Nb, and Nc.

(a)

(b)

(c)

(d)

Figure 4 

Comparisons of the responses of the plate calculated by both the analytical and the FEA methods for the air-backed case when 𝑡=0.45 ms: (a) the deflection 𝑤 of the back face of the plate calculated by the analytical method; (b) the deflection 𝑤 of the back face of the plate calculated by the FEA method; (c) the von Mises stress 𝜎𝑚 of the back face of the plate calculated by the analytical method; and (d) the von Mises stress 𝜎𝑚 of the back face of the plate calculated by the FEA method.

5.2. Validation of the Analytical Method by Comparing with Experiment Results

In this section, the analytical method is validated by comparing the acceleration of the air-backed aluminum plate calculated by this method with those obtained by the Hung et al.’s underwater explosion experiment result [26]. Hung performed the experiment within a 4×4×4 m water tank. The isotropic aluminum target plate was fixed to an empty steel casing, known as a “shock rig”, and the button of which was fixed to a steel base. In this paper, it is assumed that the plate is simply supported. The geometry and elastic properties of the plate is shown in Table 3. 1 g charge (𝑊=1 g) was located on the normal line through the center of the plate. An accelerometer was placed in the center of the plate and operated at a sampling rate of 2.0 MHz. Four tests, with different standoff distances, were carried out, although only the case with a standoff distance of 0.7 m (𝑆=0.7 m) was used for the comparisons made here. The water density is 𝜌_𝑤=1000 kg/m³, and the shock wave (acoustic) velocity in the water is 𝑐_𝑤=1500 m/s.

Table 3 

Geometry and elastic parameters of the aluminum plate.

Figure 5 compares the acceleration time histories at the center of the plate between analytical method and experiment [26]. Obviously, the time histories by the two methods agree with each other closely in the overall trend. There are some differences between the two time history curves. The acceleration peak value of analytical method is 25.6 percent greater than that of experiment. The growth can be attributed to the fact that the pressure precursor is ignored in the incident wave profile. It is understandable since the pressure precursor significantly decreases the peak of incident pressure according to [27]. In addition, there are some high-frequency vibrations in the experiment record, which are missing in the curve by the analytical method. The difference may be caused by the FSI model formulated by Taylor used here which ignores the bubble impulsive pressure. In general, the analytical method proposed in this paper and its FSI model have been validated.

Figure 5 

Comparisons of the acceleration time histories at the center of the plate between analytical method and experiment [26].

5.3. The Influence of the Fluid-Solid Interaction and the Thickness

In this section, the effects of the FSI and the thickness are investigated by the analytical method. Five different thicknesses (ℎ=0.05 m, 0.10 m, 0.15 m, 0.20 m, 0.25 m, and 0.30 m) and three cases, including the non-FSI, air-backed, and water-backed cases, are considered. The elastic properties, explosive characteristics, and geometric parameters, except for the thickness, are listed in Tables 1 and 2. In Figure 6, the extreme values of the response-time histories of 𝑤, 𝜎_𝑥𝑥, and 𝜎_𝑥𝑧 of the different thicknesses and the FSI cases are compared. The thickness and FSI effects significantly influence the reaction of the laminated plate.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 6 

Comparisons of the responses of the plate for five different thicknesses: (a) the time history of the deflection 𝑤 for the plate at 𝑥=𝑎/2,𝑦=𝑏/2, and 𝑧=ℎ/2 for the non-FSI case; (b) the time history of the deflection 𝑤 for the plate at 𝑥=𝑎/2,𝑦=𝑏/2, and 𝑧=ℎ/2 for the air-backed case; (c) the time history of the deflection 𝑤 for the plate at 𝑥=𝑎/2,𝑦=𝑏/2, and 𝑧=ℎ/2 for the water-backed case; (d) the time history of the membrane normal stress 𝜎𝑥𝑥 for the plate at 𝑥=𝑎/2, 𝑦=𝑏/2, and 𝑧=ℎ/2 for the non-FSI case; (e) the time history of the membrane normal stress 𝜎𝑥𝑥 for the plate at 𝑥=𝑎/2,𝑦=𝑏/2, and 𝑧=ℎ/2 for the air-backed case; (f) the time history of the membrane normal stress 𝜎𝑥𝑥 for the plate at 𝑥=𝑎/2,𝑦=𝑏/2, and 𝑧=ℎ/2 for the water-backed case; (g) the time history of the transverse shear stress 𝜎𝑥𝑧 for the plate at 𝑥=0,𝑦=0.5, and 𝑧=0 for the non-FSI case; (h) the time history of the transverse shear stress 𝜎𝑥𝑧 for the plate at 𝑥=0,𝑦=0.5, and 𝑧=0 for the air-backed case; and (i) the time history of the transverse shear stress 𝜎𝑥𝑧 for the plate at 𝑥=0,𝑦=0.5, and 𝑧=0 for the water-backed case.

As shown in Figures 6(a), 6(d), and 6(g), all of the extreme values of the response-time histories for the non-FSI case decrease as the thickness increases. As shown in Figures 6(b), 6(c), and 6(e), the deflection, 𝑤, and the membrane normal stress 𝜎_𝑥𝑥 in the air-backed and water-backed cases also decrease as the thickness increases. However, as shown in Figures 6(f), 6(h), and 6(i), the peak values of the response-time histories for the case of ℎ=0.05 m are less than those of ℎ=0.10 m.

6. Conclusions

It is in the interest of designers to know the transient state of stress in advance that develops in a plate during an underwater explosion. However, to the best of the authors’ knowledge, the analytical method in the relevant literature for the dynamic response of plates to underwater explosion does not consider the FSI differences between the different positions on front and back faces of plate. The purpose of this paper is to fill this gap. Involving the planar wave assumption, an analytical method was proposed based on the FSI model formulated by Taylor [15], the Mindlin plate theory, Laplace transform, and its inversion to model the elastic dynamic response of a plate subjected to an underwater explosion. This new analytical method builds upon the Reissner-Mindlin plate theory and includes the shear deformation and the FSI (fluid-solid interaction) effect. Additionally, the new method is three-dimensional, considers the moment and the membrane stress in the plate, and examines the full profile of possibilities, such as the non-FSI, the air-backed, and water-backed cases.

The response-time histories and the response distributions on the plate in terms of displacements and stresses between the analytical method and the FEA method were compared to validate the analytical method, and good agreement was found. Subsequently, the simplified method for fluid-solid interaction as well as the analytical method are validated by comparison of the acceleration at the center of an air-backed plate between analytical method and the experiment performed by Hung et al. [26].

The thickness and FSI effects significantly influence the reaction of the laminated plate. All of the extreme values of the response-time histories for the non-FSI case decrease as the thickness increases. However, considering the FSI effects for the air-backed and water-backed cases, some peak values of the response-time histories for the ℎ=0.05 m case are less than those of ℎ=0.10 m.

The inclusion of FSI effects in designs for plates is vital for a full understanding of the dynamic response, and the results can be used as benchmark solutions in further research. In addition, several types of fluid-solid interaction effects during UNDEX near the plate are coupled in the present method, including Taylor’s FSI effects, bending-stretching effects, and simply boundary effects [12]; the respective effect of each type is not presented here and will be discussed elsewhere.

Acknowledgments

The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (nos. 51209185, 51079127, 51179171, and 51279180), China Postdoctoral Science Foundation (no. 2013M531462), and the Postdoctoral Science Foundation of Zhejiang Province (no. Bsh1202057). Special thanks should go to the Ansys and Mathematica software used in this work.