Structural analysis of a submerged fluid-filled cylindrical shell subjected to a shock wave

Journal of Fluids and Structures 90 (2019) 450–477

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Structural analysis of a submerged fluid-filled cylindrical shell subjected to a shock wave S. Iakovlev Department of Engineering Mathematics and Internetworking, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2

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Article history: Received 24 October 2018 Received in revised form 27 May 2019 Accepted 5 June 2019 Available online 21 October 2019

a b s t r a c t Three-dimensional structural dynamics of a submerged fluid-filled circular cylindrical shell subjected to an external shock wave is addressed for the scenario of different internal and external fluids. A semi-analytical model based on the use of the classical apparatus of mathematical physics is employed, and the focus of the study is on understanding how the peak values of the compressive and tensile stress evolve when the acoustic properties of the internal fluid change. It is demonstrated that the peak tensile stress can be very sensitive to such changes, while the respective response of the peak compressive stress is quasi-linear. It is also shown that for the present system, it is not always possible to rely on the parametric studies based on the use of simplified, computationally cheaper two-dimensional models even when conservatives estimates of the peak stress are sought because the peak stress values predicted by such models can be both higher and lower than the values predicted by the more realistic, fully threedimensional models. The dominance of the transverse stress over the longitudinal one is established for all scenarios considered. The spatial and temporal locations of the stress peaks are analyzed as well, and the practical implications of the respective findings are highlighted. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The present study is devoted to the analysis of the dynamics of the stress state of a submerged fluid-filled cylindrical shell subjected to an external spherical shock wave with the three-dimensionality of the system being fully preserved by the employed mathematical model. It is further assumed that the internal and external fluids can have different properties. The motivation for the study is as follows. It has been demonstrated (Iakovlev et al., 2011) that when a submerged cylindrical shell subjected to an external shock wave is filled with a fluid with the properties that differ from those of the external fluid, the peak structural stress in the shell can differ very dramatically from that observed for the case of two identical fluids inside and outside the shell; the difference was observed to be particularly striking for the peak tensile stress. That study, however, was carried out only for the two-dimensional simplification of the system, thus somewhat limiting the utility of the reported findings as far as their direct industrial application is concerned — as has been recently demonstrated for a system of a similar type (Iakovlev, 2018), the three-dimensionality of the model plays a very significant role in determining the correct values of the peak stress observed in the system even when the qualitative behavior of the peak stress is still quite adequately captured by the two-dimensional model. Thus, a more realistic, fully three-dimensional investigation would be very welcome. E-mail address: [email protected]. https://doi.org/10.1016/j.jfluidstructs.2019.06.001 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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Nomenclature ci ce cs h0 ps pα p0 pd pi pr r r0 R0 SR t u∗

v∗ w∗ X

θ λ ν i ξmn e ξmn ρi ρe ρs ϱ ∗ σ11 ∗ σ22 τ φ

sound speed in the internal fluid sound speed in the external fluid sound speed in the shell material thickness of the shell total pressure on the shell surface peak incident pressure incident pressure diffraction pressure internal pressure radiation pressure dimensionless radial coordinate of the cylindrical coordinate system radius of the shell radial distance to the source of the incident shock waves stand-off of the source of the incident shock waves dimensionless time longitudinal displacement of the middle surface of the shell transverse displacement of the middle surface of the shell normal displacement of the middle surface of the shell axial coordinate of the cylindrical coordinate system angular coordinate of the cylindrical coordinate system exponential decay rate Poisson’s ratio internal response function external response function density of the internal fluid density of the external fluid density of the shell material dimensional radial coordinate of the cylindrical coordinate system longitudinal stress Transverse stress dimensional time fluid velocity potential with the subscripts bearing the same meaning as for the pressure components

Other variables are defined in the text.

The goal of the present study, therefore, is to consider a three-dimensional generalization of the two-dimensional analysis presented in Iakovlev et al. (2011), and to do so in the spirit of the study summarized in Iakovlev (2018). More specifically, it is of particular interest to explicitly visualize and analyze the changes that are brought to the dynamics of the stress state by the changing properties of the internal fluid, and then to understand how these changes affect the overall peak stress observed in the system, as well as to determine how the evolution of the peak stress is different from that seen when the simplified two-dimensional model is used. These insights would be, as far as we are aware, novel, and would enhance the current understanding of the complex physics of the shell–shock interaction; at the same time, the mathematical apparatus that we intend to use has, for most part, been already established in our previous work, thus we claim no major novelty in that regard. From the practical point of view, carrying out the proposed analysis will enable better understanding of the shock response of a rather wide variety of industrial systems, including, but certainly not limited to, underwater oil and other fluid-transporting pipelines, underwater storage tanks, heat exchange systems and fuel tanks of various configurations. As far as the place of the present study in the existing literature is concerned, it belongs, as did the two studies that inspired it, to the rather vast area of the shock–structure interaction research that addresses systems or/and interaction regimes that are more complex than the classical scenario of a single, stand-alone structure subjected to a single shock wave. It would not be appropriate for a non-review paper to attempt an overview, however brief, of all efforts that have been undertaken in addressing such complex situations. Instead, we mention three focus areas that appear to be most

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relevant to the present study while highlighting how what we intend to accomplish here is different from what has already been done. The first focus area concerns with analyzing the shock response of systems that involve a certain degree of material complexity such as the use of composites (e.g. Batra and Hassan, 2007; Chen et al., 2013; Panahi et al., 2011; Wang et al., 2013; Fan et al., 2016) or that of various coatings of the shock-responding structure (e.g. Yin et al., 2016; Xiao et al., 2014). Most of the published work, however, addresses structural material complexity, not the multi-fluid complexity we intend to consider here. There exist studies devoted to the analysis of the shock response of systems where fluid is included in a double-layer structural configuration (e.g. Schiffer and Tagarielli, 2014; Liu et al., 2017), but none of them consider the effect of varying the properties of the fluids comprising the system, while assessing such an effect is at the core of the present study. The second focus area addresses the shock response of systems undergoing, broadly put, a complex loading scenario, with such complex phenomena considered as the shock-induced cavitation or/and the interaction with the explosion bubble (e.g. Xie et al., 2008; Wang et al., 2014; Brett and Yiannakopolous, 2008; Klaseboer et al., 2005; Lee et al., 2011; Ming et al., 2016). However, to the best of our knowledge, none of the studies of this group consider the effects of the presence in the system of two or more different fluids with varying properties. The third group can be said to be comprised of studies devoted to analyzing the effect of various forms of geometrical complexity present in the system. To this group belong the studies relying on fully realistic, high-fidelity geometrical modeling aimed at assessing the shock response of certain rather specific industrial configurations (e.g. Zhang et al., 2014; Kim and Shin, 2008; Guo et al., 2017) as well as studies addressing the effect of various reflective surfaces present within the considered system (e.g. Walters et al., 2013; Wardlaw and Luton, 2000; Oakley et al., 2001; Jappinen and Vehvilainen, 2006). Again, none of the works of this group seem to focus on the analysis on the changing properties of the fluids comprising the system. Thus, it appears that, to date, there has been no research effort specifically devoted to determining how changes of the properties of different fluids comprising a shock-responding thin-walled system affect the stress state of the system. The study we summarize here attempts to partially fill this gap. 2. Mathematical formulation We consider a circular cylindrical shell of thickness h0 and with the radius of its middle surface r0 . We assume that h0 ≪ r0 and that the shell displacements are small compared to its thickness, and we also assume that the Kirchhoff–Love hypothesis holds true. The density of and the sound speed in the shell material are ρs and cs , respectively. The longitudinal, transverse, and normal displacements of the middle surface of the shell are u∗ , v ∗ and w ∗ . The shell is filled with and submerged into linearly compressible, irrotational and inviscid fluid, and it is assumed that the internal and external fluids have different properties, with the ρi and ci being the density of and the sound speed in the internal fluid, and ρe and ce being the density of and the sound speed in the external fluid. The cylindrical coordinate system (ϱ, θ , X ) based on the axis of the shell is employed, Fig. 1. The fluids are governed by the wave equations for the internal and external fluid velocity potentials, φi and φe , respectively,

∇ 2 φi =

1 ∂ 2 φi

(1)

ci2 ∂τ 2

and

∇ 2 φe =

1 ∂ 2 φe ce2 ∂τ 2

.

(2)

While the internal potential has only one component, φi , the external one is comprised of three components,

φe = φ0 + φd + φr ,

(3)

where φ0 is the incident potential, φd is the diffraction potential, and φr is the radiation potential. The corresponding pressure components are pi , pe , p0 , pd and pr , with pe = p0 + pd + pr .

(4)

The total pressure acting on the shell surface, ps , is then ps = (p0 + pd + pr − pi )|ϱ=r0 .

(5)

The shell is governed by the equations of the Kirchhoff–Love linear shell theory (Ugural, 1981),

∂ 2 u∗ 1 − ν ∂ 2 u∗ 1 + ν ∂ 2 v∗ ν ∂w∗ + + − = 0, 2 2 2 ∂X 2r0 ∂ X ∂θ r0 ∂ X 2r0 ∂θ { 3 ∗ } 1 + ν ∂ 2 u∗ 1 − ν ∂v ∗ 1 ∂ 2 v∗ 1 ∂w ∗ ∂ w 1 ∂ 3 w∗ ∂ 2 v∗ 1 ∂ 2 v∗ 2 + + − + k + + (1 − ν ) + = 0, 0 2r0 ∂ X ∂θ 2 ∂X2 ∂ X 2 ∂θ ∂X2 r0 ∂θ 2 r02 ∂θ 2 r02 ∂θ r02 ∂θ 3

(6) (7)

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Fig. 1. Geometry of the problem.

ν ∂ u∗ 1 ∂v ∗ w∗ + 2 − 2 − k20 r0 ∂ X r0 ∂θ r0

(

∂ 4 w∗ 2 ∂ 4 w∗ 1 ∂ 4 w∗ ∂ 3 v∗ 1 ∂ 3 v∗ + + − − ∂X4 ∂ X 2 ∂θ r02 ∂ X 2 ∂θ 2 r04 ∂θ 4 r02 ∂θ 3

) =

1 ∂ 2 w∗ cs2 ∂τ 2

− χ ps ,

(8)

where ν is Poisson’s ratio of the shell material, k20 =

h20 12r02

and χ =

1 h0 ρs cs2

.

(9)

In Eqs. (6)–(8), all the bending stiffness terms are fully retained — although neglecting some of these terms is possible (Junger and Feit, 1972), we did not see any particularly significant benefit in doing so in this study.

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The potential components satisfy the boundary conditions on the interface,

⏐ ∂φi ⏐⏐ ∂w∗ =− ⏐ ∂ϱ ϱ=r0 ∂τ

⏐ ∂φr ⏐⏐ ∂w∗ , =− ⏐ ∂ϱ ϱ=r0 ∂τ

and

⏐ ⏐ ∂φd ⏐⏐ ∂φ0 ⏐⏐ = − , ∂ϱ ⏐ϱ=r0 ∂ϱ ⏐ϱ=r0

(10)

the decay conditions at the infinity,

φd −→ 0 and φr −→ 0 when ϱ → ∞,

(11)

the condition at the center of the fluid domain,

|φi | |ϱ=0 < ∞,

(12)

and the zero initial conditions. As for the axial boundary conditions, we make use of the fact that all the perturbations in the shell propagate with the finite velocity given by cs . That is, if we choose the parameter L∗ such that all the shell deformations are confined to the region X ∈ (−L∗ , L∗ ) during the entire time period that is of interest to us, then any of the standard boundary conditions can be set at X = ±L∗ simply because those regions are not yet affected by the loading (a more detailed discussion of the matter can be found in Iakovlev, 2018). For the present system, the following choice appeared to be particularly beneficial:

⏐ ∂ u∗ ⏐⏐ = 0, ∂ X ⏐X =±L∗

v

⏐

∗⏐

X =±L∗

= 0,

w

⏐

∗⏐

X =±L∗

= 0, and

⏐ ∂ 2 w ∗ ⏐⏐ = 0. ∂ X 2 ⏐X =±L∗

The value of L∗ was chosen (Iakovlev, 2018) as cs L∗ = 10 r0 + δ, ce

(13)

(14)

where δ is an arbitrary parameter with any δ > 0 being suitable, which assumes that the maximum value of the time we consider corresponds to the instant when the incident front has moved over the distance equal to ten radii of the shell from the point if its initial contact with the structure. The boundary conditions for the fluid velocity potentials are set using the same conceptual framework, and using the same interval (−L∗ , L∗ ) due to the fact that in all scenarios of practical interest, the sound speed in the fluids would not be exceeding that in the shell:

⏐ ⏐ ∂ 2 φr ⏐⏐ ∂ 2 φd ⏐⏐ = 0, φr |X =±L∗ = 0, = 0, φd |X =±L∗ = 0, ∂ X 2 ⏐X =±L∗ ∂ X 2 ⏐X =±L∗ ⏐ ∂ 2 φi ⏐⏐ φi |X =±L∗ = 0, and = 0. ∂X2 ⏐ ∗

(15) (16)

X =±L

It should, therefore, be noted that the term ‘submerged’ in the present context should be taken in the same sense as in all the works on the shell–fluid interaction where an infinitely long cylindrical shell is considered, be it a 2D or a 3D model, that is, as having fluid in contact only with its external surface. This clarification needed to be made to avoid confusion over the uncertainty regarding the potential perceived existence of the contact between the introduced ‘ends’ of the shell at x = ±L∗ and the fluids (of which there is none) - in other words, to clarify that the shell is not ‘closed’ by any end caps at x = ±L∗ , and that that boundary is purely fictitious and has no impact on the physics of the problem. The shell is subjected to a shock wave originated at the point located at the distance R0 from the axis of the shell, and an exponential decay of the pressure behind the front of the wave is assumed. The fluid velocity potential φ0 in such a wave is given by (Cole, 1948; Iakovlev, 2006)

φ0 = −

} λpα SR { −(τ −ce−1 (R∗ −SR ))λ−1 e − 1 H(τ − ce−1 (R∗ − SR )), ρe R∗

(17)

R20 + X 2 + ϱ2 − 2R0 ϱ cos θ,

(18)

where R∗ =

√

pα is the pressure in the front of the wave when it first contacts the shell, λ is the exponential decay constant, SR = R0 − r0 is the stand-off of the source of the wave, and H is the Heaviside unit step function. It should be noted that the chosen linear shell model implies that only weak shock waves can be addressed in the present study, otherwise the induced displacements of the structural surface would be too large, and a non-linear shell model would be necessary (e.g. Amabili, 2018). When the parameters of the loading considered in Section 5 were chosen, this limitation was certainly kept in mind; moreover, after the simulations were completed, the peak displacements of the shell surface were analyzed for all the addressed scenarios in order to ensure they are well within the limits imposed by the adopted linear theory (see Section 5). As for the stress state, it has been demonstrated (Iakovlev, 2004) that for the shell thickness and for the type of the incident loading we consider here, the stress state of the shell is dominated by the membrane components, with the

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bending components providing only a relatively small contribution to the total stress and only in the regions that are ∗ and very localized. Thus, we only consider the membrane stress components and assume that the transverse stress σ22 ∗ the longitudinal stress σ11 are given by

σ22 ≈ ∗

(

E 1 − ν2

1 ∂v

∂u − w+ν r0 ∂θ r0 ∂x 1

) (19)

and ∗ ≈ σ11

(

E 1 − ν2

) ∂u ν ∂v ν + − w , ∂x r0 ∂θ r0

(20)

where E is Young’s modulus of the shell material. We consider a dimensionless formulation of the problem normalizing all variables to r0 , ce , and ρe . A macron normally distinguishes a dimensionless variable from its dimensional counterpart, with the exception of the time t = τ ci r0−1 , the radial coordinate r = ϱr0−1 , the axial coordinate x = Xr0−1 , the axial ∗ −1 −2 ∗ −1 −2 boundary condition parameter L = L∗ r0−1 , the stresses σ11 = σ11 ρe ce and σ22 = σ22 ρe ce , and the shell displacements, ∗ −1 ∗ −1 ∗ −1 w = w r0 , v = v r0 and u = u r0 . 3. Fluid dynamics The solution methodology we use is essentially the same as the one employed in our earlier studies (Iakovlev, 2004, 2018), thus we only summarize it here. Specifically, the Laplace transform is first applied to the dimensionless wave equations written in the cylindrical coordinates to arrive at

¯e ¯e ¯e ¯e 1 ∂Φ ∂ 2Φ 1 ∂ 2Φ ∂ 2Φ ¯e = 0 + + + 2 − s2 Φ 2 2 ∂r r ∂r ∂x r ∂θ 2

(21)

and

¯i ¯i ¯i ¯i ∂ 2Φ 1 ∂Φ ∂ 2Φ 1 ∂ 2Φ ¯ i = 0, + + + − s2 α 2 Φ (22) 2 2 2 ∂r r ∂r ∂x r ∂θ 2 ¯ e and Φ ¯ i are the Laplace transforms of φ¯ e and φ¯ i , respectively, α = ce /ci , and s is the transform variable. Then, the where Φ spatial variables in (21) and (22) are separated to arrive at the general solutions of the equations satisfying the necessary boundary conditions as

¯ e = Amn Kn (r βm (s)) cos mx ¯ cos nθ, Φ

m = 0, 1, . . . , n = 0, 1, . . .

(23)

¯ i = Cmn In (r αβm (s)) cos mx ¯ cos nθ, Φ

m = 0, 1, . . . , n = 0, 1, . . . ,

(24)

and

where (2m + 1)π

, √2L ¯ 2 + s2 , βm (s) = m ¯ = m

(25) (26)

In and Kn are the modified Bessel functions of the first and second kind, respectively, of order n, and Amn and Cmn are arbitrary functions of s. Assuming that the normal displacement of the shell and the normal incident velocity on the shell surface are also represented as

w=

∞ ∑ ∞ ∑

¯ cos nθ wmn cos mx

(27)

m=0 n=0

and

⏐ ∞ ∑ ∞ ∑ ∂ φ¯ 0 ⏐⏐ ¯ cos nθ, = bmn cos mx ⏐ ∂ r r =1

(28)

m=0 n=0

and imposing the remaining boundary conditions, the Laplace transforms of the potential components can be obtained as follows (from here on, it is assumed that we only consider the ‘surface’ version of all the functions concerned, that is, that r = 1): d e ¯ mn ¯ cos nθ, Φ = Bmn (s) Ξmn (s) cos mx

(29)

r e ¯ mn ¯ cos nθ, Φ = sWmn (s) Ξmn (s) cos mx

(30)

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S. Iakovlev / Journal of Fluids and Structures 90 (2019) 450–477

and i i ¯ mn ¯ cos nθ, Φ = −sWmn (s) Ξmn (α s) cos mx

(31)

where Wmn and Bmn are the Laplace transforms of wmn and bmn , respectively, and ξ response functions of the problem with the Laplace transforms given by K(βm (s)) βm (s) K′ (βm (s))

e (s) = − Ξmn

e mn

and ξ

i mn

are the external and internal

(32)

and i (s) = Ξmn

I(βm (s))

βm (s) I′ (βm (s))

,

(33)

respectively, where the prime denotes the derivative of the function with respect to its argument. The present threedimensional response functions can be expressed (Iakovlev, 2004) in terms of their two-dimensional counterparts ξne (Iakovlev, 2008b) and ξni (Iakovlev, 2006) as t

∫

e ¯ (t) = ξne (t) − m ξmn

√ ¯ η ) dη ξne ( t 2 − η2 )J1 (m

(34)

√ ¯ η) dη, ξni ( t 2 − η2 )J1 (m

(35)

0

and i ¯ ξmn (t) = ξni (t) − m

∫

t

0

where J1 is the Bessel function of the first order. The response functions represent all the essential physics of the scattering and radiation by the shell and were discussed in their present three-dimensional form in Iakovlev (2004); further insights into the relation between the response functions and the fundamental physics of the interaction can be gained from the detailed discussions of their two-dimensional counterparts found in Iakovlev (2006) and Iakovlev (2008b). Then, the pressure components can be obtained as p¯ d =

∞ ∑ ∞ ∑

¯ cos nθ, p¯ dmn cos mx

(36)

¯ cos nθ, p¯ rmn cos mx

(37)

¯ cos nθ, p¯ imn cos mx

(38)

m=0 n=0

p¯ r =

∞ ∑ ∞ ∑ m=0 n=0

and ∞ ∑ ∞ ∑

p¯ i =

m=0 n=0

where pdmn

¯

t

∫

bmn (η)

= −bmn (t) −

e dξmn

0

p¯ rmn = −

d2 w

t

∫

dη 2

0

dη

(t − η) dη,

e (η) ξmn (t − η) dη

(39)

(40)

and pimn

¯

ρi ci = ρe ce

t

∫ 0

d2 w dη 2

(η ) ξ

i mn

(

ci ce

(t − η)

)

dη,

(41)

and the total pressure on the shell surface is given by ∞ ∑ ∞ ∑

¯ cos nθ, p¯ mn cos mx

(42)

p¯ mn = p¯ 0mn + p¯ dmn + p¯ rmn − p¯ imn ,

(43)

p¯ =

m=0 n=0

where

with p¯ 0 =

∞ ∑ ∞ ∑ m=0 n=0

¯ cos nθ. p¯ 0mn cos mx

(44)

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4. Structural dynamics The displacements are also sought in the form of the double series, u(x, θ , t) =

∞ ∑ ∞ ∑

¯ cos nθ, umn (t) sin mx

(45)

¯ sin nθ, vmn (t) cos mx

(46)

m=0 n=0

v (x, θ , t) =

∞ ∑ ∞ ∑ m=0 n=0

w(x, θ , t) =

∞ ∑ ∞ ∑

¯ cos nθ, wmn (t) cos mx

(47)

m=0 n=0

which, for each pair (m, n), yields the following system of ordinary integro-differential equations for the modal coefficients umn , vmn , and wmn :

γ2 γ2 γ

d2 umn dt 2 d2 vmn

dt 2 2 d wmn 2

11 12 13 + cmn umn + cmn vmn + cmn wmn = 0,

(48)

21 22 23 + cmn umn + cmn vmn + cmn wmn = 0,

(49)

31 32 33 + cmn umn + cmn vmn + cmn wmn = χ¯ p¯ mn , dt 2 where γ = ce /cs and 11 cmn = 21 cmn =

1−ν 2

1+ν 2

23 32 cmn = cmn

¯ 2, n2 + m ¯ , mn

12 cmn =

1+ν

1−ν

2

¯ , mn

13 31 ¯, cmn = cmn = −ν m

¯ 2 + n2 + k20 (n2 + 2(1 − ν )m ¯ 2 ), m 2 33 ¯ 2 ), and cmn ¯ 2 + n2 )2 . = −n + k20 (n3 + (2 − ν )nm = 1 + k20 (m 22 cmn =

(50)

(51)

The system (48)–(50) was approached numerically using an explicit finite difference scheme (e.g., Iakovlev, 2009), and the time step of 0.00125 was determined to be sufficient to ensure convergence in all cases of interest. As far as the series convergence is concerned, 201 terms for the axial coordinate and 81 terms for the angular one always ensured the convergence. As for the validation of the developed model, due to the mentioned lack of published work addressing the system in question it was not possible to carry out a direct validation. Nevertheless, we feel quite confident in recommending the proposed solution due to the following two reasons. First, the solution is based on the 2D framework that has been extensively and successfully validated in our previous work (Iakovlev, 2006, 2008a; Iakovlev et al., 2013, 2014) by comparing the results of the numerical simulations with the data produced by several independent experimental investigations. Second, the 3D solution itself has also been successfully validated using the fact that all the perturbations in both the shell and the fluids propagate with known constant velocities. 5. Results and discussion 5.1. General considerations We consider a steel shell of thickness h0 = 0.01 m and radius r0 = 1.00 m, cs = 5000 m/s, ρs = 7800 kg/m3 , and ν = 0.3, submerged in water, ce = 1400 m/s and ρe = 1000 kg/m3 . The sound speed in the internal fluid, ci , varies. The density of the internal fluid is assumed to be the same in all cases, namely ρi = 1000 kg/m3 - although the respective properties may not always exactly correspond to any existing fluid, such a choice appears to be well-justified for the purposes of the present study (Iakovlev et al., 2011). The incident wave is assumed to have the rate of exponential decay λ = 0.1314 ms and the pressure in the front at the moment of the initial contact with the shell pα = 250 kPa. It is further assumed that the shock wave is originated at the source located at the distance of R0 = 5r0 from the axis of the shell (the shock wave standoff of SR = 4r0 ). Since it has been shown (e.g. Iakovlev, 2009; Iakovlev et al., 2011) that the single most important parameter that drives the evolution of the fluid and structural dynamics is the ratio of the sound speeds in the internal and external fluids, ci (52) ζ = , ce our analysis will focus on providing insights into how the change of ζ affects various aspects of the system’s structural dynamics.

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Fig. 2. Stress state of the shell for the case of ζ = 1.00 at t = 0.40, (a), t = 0.80, (b), t = 1.10, (c), t = 2.10, (d), t = 2.20, (e), and t = 2.60, (f); the left column shows the front view of the shell, and the right column shows the back view of the shell.

S. Iakovlev / Journal of Fluids and Structures 90 (2019) 450–477

Fig. 2. (continued).

459

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Fig. 3. Stress state of the shell for the case of ζ = 0.50 at t = 0.40, (a), t = 1.20, (b), t = 2.10, (c), t = 3.00, (d), t = 4.00, (e), t = 4.10, (f), t = 4.20, (g), t = 4.40, (h), and t = 5.60, (i); the left column shows the front view of the shell, and the right column shows the back view of the shell.

S. Iakovlev / Journal of Fluids and Structures 90 (2019) 450–477

Fig. 3. (continued).

461

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Fig. 3. (continued).

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As for the relevant values of ci , we considered the variation from 300 to 1400 m/s (in keeping with the parent study, Iakovlev et al., 2011, only the fluids with ci ≤ ce were addressed). These values appear to correspond to the majority of the fluids that are of practical interest in the present context (Selfridge, 1985), yielding the range of [0.214, 1] for ζ . The lower values are meant to account for various rather exotic scenarios (certain liquified gases), while the more common internal fluids would normally yield the values exceeding 0.5, with gasoline, petroleum, liquid oxygen and liquid nitrogen producing the values of 0.89, 0.92, 0.64 and 0.61, respectively. Finally, we note that in order to ensure that the choice of the employed linear shell theory was well-justified for the considered loading, the maximum normal displacement of the shell surface was analyzed in all considered cases. It was established that it never exceeded 1% of the thickness of the shell — an outcome that certainly alleviates any concerns one might have about the suitability of the employed linear shell theory for the considered loading intensity. 5.2. 3D stress dynamics Although the ultimate goal of this study is the analysis of the peak structural stress, it is first necessary to gain some understanding of how the three-dimensional dynamic stress fields evolve in response to the changes of ζ . To that end, we present summary 3D images illustrating the dynamics of the transverse stress for the three very different scenarios, namely ζ = 1.00 (identical fluids inside and outside), ζ = 0.50 (the internal fluids has a significantly lower sound speed) and ζ = 0.214 (the internal sound speed being at the lower limit of what is practically meaningful), Figs. 2–4, respectively. We note that for now, we focus on the transverse stress only because it has been observed to be the dominant stress in certain similar systems (e.g. Iakovlev, 2004, 2018); later on, however, we will also examine the longitudinal stress. We emphasize that the frames included in these three figures are primarily aimed at providing insight into how the peak stress, both compressive and tensile, is reached, not at giving a detailed overall summary of the evolution of the stress state — for the latter purpose, a considerably higher number of frames would be necessary. The colors in all the images were assigned so that the blue corresponds to the highest stress and the red to the lowest; all frames for a particular ζ were scaled identically, with the global extrema of the stress dictating the color distribution. Finally, we note that although simulating the three-dimensional fluid dynamics of the system is outside of the scope of the present study, the hydrodynamic field images found in the related two-dimensional investigations (e.g. Iakovlev, 2009; Iakovlev et al., 2011) could serve as quite an informative complement to the structural images presented here. The case of ζ = 1.00 (ci = 1400 m/s) has been discussed extensively in our earlier studies (e.g. Iakovlev, 2006, 2007, 2009), and can be considered rather well-understood by now, thus it only needs a brief summary. Namely, after the stress wave is originated at the instant of the initial contact of the shock wave with the shell, frames (a) of Fig. 2, it circumnavigates the shell, frames (b), with the constructive superposition of its two symmetric branches producing the peak compressive stress at the tail point at t = 1.08, frames (c). At t = 2.0, the pressure wave in the internal fluid arrives at the shell surface in the tail region and causes the high tensile stress there, frames (d), that peaks at t = 2.15 producing the global maximum of σ22 , frames (e). After that, the stress state becomes more and more complex due to the multiple superpositions of the stress waves propagating in the shell while the stress magnitude diminishes, frames (f). The case of ζ = 0.5 (ci = 700 m/s) is, expectedly, very different from the case of the identical internal and external fluids. The early stages of the stress dynamics are qualitatively very similar to the case of ζ = 1.0 - the stress wave is circumnavigating the shell after being originated in the head region, frames (a) and (b) of Fig. 3, with the peak compressive stress reached at t = 1.19 due to the same superposition of the two symmetric branches of the stress wave as before. However, unlike in the previous case, no pressure wave in the internal fluid arrives at the tail region of the shell at t = 2.0, and the stress state is continuing to be dominated by the multiple superpositions of the stress waves propagating longitudinally and transversely, frames (c) and (d). The arrival of the internal pressure wave at the tail region now occurs much later in the interaction, at t = 4.0, frames (e), with the resulting tensile peak reached at t = 4.12, frames (f) and (g). After that, the stress field becomes increasingly complex, frames (h) and (i), while the stress magnitude diminishes. The case of ζ = 0.214 (ci = 300 m/s) is, perhaps, most interesting. Although the stress dynamics during the early interaction is qualitatively very much the same as in the previous two cases, frames (a)–(c) of Fig. 4, it is quantitatively somewhat different in that the peak compressive stress is now reached in the head region (just under 6o off the head line θ = 0 but still in the mid-section x = 0), and it occurs much earlier in the interaction, at t = 0.40. After that, for a while the stress dynamics is dominated by the propagation of the stress waves in the longitudinal and transverse directions, frames (d). The most remarkable difference between the present and the other two scenarios we considered is, however, in the fact that the peak tensile stress is now reached not in the mid-section of the shell x = 0 after the arrival at the tail point of the pressure wave in the internal fluid at t = 9.33, frames (i)–(l), but at a point that is located quite far away from the mid-section. Namely, the regions of high tensile stress that develop in the shell and that are clearly detectable by t = 3.0, frames (e), become more and more pronounced, and eventually (at t = 3.9, frames (f)) the respective stress becomes so high that it exceeds the highest stress due to the reflection of the internal wave off the tail region. We note that similar tensile stress formations were present for ζ = 0.5 as well, but in that case the respective stress was not high enough to produce the global maximum. The period between these two peaks is, phenomenologically speaking, rather uninteresting, frames (g) and (h). From the practical point of view, the fact that the global tensile maximum is reached not in the mid-section of the shell has some rather important implications, and we return to this issue later.

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Fig. 4. Stress state of the shell for the case of ζ = 0.214 at t = 0.40, (a), t = 0.80, (b), t = 1.10, (c), t = 2.60, (d), t = 3.00, (e), t = 3.90, (f), t = 5.40, (g), t = 7.10, (h), t = 9.40, (i), t = 9.50, (j), t = 9.70, (k), and t = 9.80, (l); the left column shows the front view of the shell, and the right column shows the back view of the shell.

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Fig. 4. (continued).

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Fig. 4. (continued).

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Fig. 4. (continued).

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Fig. 5. Peak tensile transverse stress in the shell as a function of ζ : 3D model, red; 2D model, black.

Fig. 6. Ratio of the peak tensile transverse stress produced by the 2D model to that produced by the 3D model.

Analyzing the three image sequences, one is compelled to conclude that the dynamics of the stress state undergoes some rather dramatic phenomenological changes as ζ diminishes. Naturally, it is exactly these changes that are the chief reason for the existence of the very different values of the peak stress, both tensile and compressive, that were seen and discussed in much detail in the two-dimensional case (Iakovlev et al., 2011); this point itself does not require any further elaboration. What does, however, prompt a thorough investigation is the need to understand how the effect of the changes of ζ on the peak stress is different in the present three-dimensional case in comparison to its two-dimensional counterpart.

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5.3. Peak stress - 2D vs. 3D As was seen in the parent study (Iakovlev et al., 2011), the peak tensile stress is far less predictable than the peak compressive one, thus we start our discussion with analyzing the evolution of the tensile peak. To that end, Fig. 5 shows the comparison between the peak tensile stress attained in the 2D and 3D cases. Although qualitatively the evolutions of the stress peaks exhibit a number of similarities, quantitatively this is not at all the case, and these quantitative differences require a most careful examination. The first, and perhaps most important, conclusion one can draw is that it is not possible to state that using the 2D model would always lead to an overestimation of the peak tensile stress (a conclusion that one would be tempted to draw based on considering the case of ζ = 1.0 alone) - there are several ranges of ζ where the 3D peaks are higher, and not inconsiderably, than their 2D counterparts. To that end, Fig. 6 shows the ratio of the 2D and 3D stress peaks, and it is clear that although for some values of ζ the 2D model may overestimate the peak tensile stress by as much as 60%, for other ζ it can also underestimate the peak by more than 20% - by no means an insignificant underestimation. From the practical point of view, this finding is of considerable importance because it clearly indicates that it is not always possible to substitute the 3D parametric studies by their incomparably computationally cheaper 2D counterparts even when a preliminary, conservative estimate of the peak tensile stress is sought, and that sometimes a fully-3D analysis is necessary. What causes such a significant difference in the peak stress observed in the 2D and 3D cases? The answer is seemingly simple (but, despite this apparent simplicity, containing all kinds of potentialities for the future research efforts): The observed difference is owed to the two fundamentally different ways in which the energy of the initial impact is distributed in the system. Namely, in the 2D case, there is only one direction in which the energy can dissipate - the transverse one, with no variation observed in the axial direction. In the 3D case, in contrast, there are two directions in which the energy can dissipate, the transverse one and the axial one, with a variety of wave phenomena taking place in both directions while producing secondary and higher-order waves that, in turn, propagate (also bi-directionally) and interact with each other, with a very complex wave pattern resulting. And it is exactly this complexity that is responsible for the differences we have reported. The 2D model is, naturally, incapable of capturing any of these ‘bi-directional’ phenomena, and this inability constitutes its main shortcoming. What, then, are the practical situations where the 2D model is sufficient, and what are the ones where it is essential to use the 3D model? Generally speaking, a fully-3D investigation is necessary for any loading where the threedimensionality is pronounced, the most obvious example being an explosion originated at the source located in a relative proximity of the structure (keeping in mind, of course, that the present model is not suitable for the scenario of a very close explosion, Iakovlev, 2006). On the other hand, when the front of the loading is more or less plane, as would be the case when the effects of an explosion with a distant source are considered, the 2D model would be sufficient (pressure pulses generated by a sudden movement of a planar surface located in the relative proximity of the structure would also fall into this category). Another practically important outcome is the observation of the rather dramatic change that the peak tensile stress undergoes in the proximity of ζ = 0.8. Namely, increasing ζ by less than 3% (from 0.786 to 0.807) produces as significant an increase of the peak tensile stress as 35%. The existence of such an effect most certainly is of interest to the practitioner — although controlling the properties of the fluids comprising the system is often not an option, the prospects of the 35% decrease of the peak tensile stress is too important a possibility to be excluded from the pre-design considerations. We note that this phenomenon was observed and discussed in detail in the parent 2D study as well (Iakovlev et al., 2011), but there it was quantitatively different. We further note that there are two other ranges of ζ where sudden, although less significant, changes of the peak stress are seen, one around ζ = 0.45, and the other around ζ = 0.33. What mechanism is behind these sudden changes of the peak tensile stress, particularly (and most importantly) the one seen at ζ ≈ 0.8? The answer to this question deserves a particularly close attention because the mechanism it outlines is very characteristic of the physics of the systems of the type considered here (we note that although, ideally speaking, it would be appropriate to illustrate the following discussion with two sequences of the relevant 3D images of the stress state similar to Figs. 2–4, a decision was made against doing so due to the already high number of images in the paper; we do not feel that this decision was in any way detrimental to the value of the presented analysis since a general idea of what is described below can well be obtained from the examination of the already included figures, in particular of the frames (d) of Fig. 2 and frames (c) of Fig. 3). Specifically, there are two physically very different waves that are responsible for the sudden change in question — the already discussed elastic wave propagating in the shell and the hydrodynamic wave propagating in the internal fluid. As we have seen, the first tensile peak observed in the tail region is caused by the hydrodynamic wave arriving at that region after it transverses the internal fluid volume once; as ζ decreases, this happens increasingly later in the interaction (e.g., at t = 2.35 for ζ = 0.85 compared to t = 2.86 for ζ = 0.70). Rather independently of this, the stress wave is circumnavigating the shell, with various parts of its two symmetric branches superposing constructively at the head and tail points with some regularity and causing higher compressive stress there; we emphasize that for the given shell material properties, these superpositions always occur at the same time, regardless of the (varying) properties of the internal fluid. Thus, the tensile peak in question continues to shift in time as ζ decreases, while the compressive peak always takes place at the same time, with both peaks occurring in the same region of the shell.

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Fig. 7. Peak compressive transverse stress in the shell as a function of ζ : 3D model, red; 2D model, black.

Having clarified this, we can now clearly see the reasons for the sudden drop of the tensile peak value — when ζ is noticeably greater than 0.80, the peak occurs before the respective branches of the stress wave reach the tail point and superpose there. As ζ decreases, the tensile peak occurs later and later, and it starts to coincide with the compressive peak caused by the stress wave. This simultaneous occurrence of the stress of the opposite sign very considerably reduces the high tensile stress observed in the region, even if the mechanism that induces this high tensile stress remains the same for all ζ in question. The other two sudden changes seen for the lower values of ζ are caused by the mechanism of much the same nature. The analysis of the peak compressive stress presents one with far fewer difficulties, as is evident from Fig. 7 and the accompanying Fig. 8 showing the ratio of the two peaks — not only qualitatively the evolutions of the peaks are nearly identical, the 3D peak is almost the exact multiple of the 2D one, with the respective coefficient varying between 1.60 and 1.66. The similarity is so striking that one is tempted to propose as simple a solution as relying on the 2D model with the subsequent scaling of the results it produces by a factor that is the same for all ζ . 5.4. σ11 vs. σ22 - is σ22 always dominant? In all the cases we have considered so far (e.g. Iakovlev, 2004, 2018), the transverse stress σ22 was always dominant. However, in the light of the rather dramatic differences in the dynamics of the stress state we have seen for various values of ζ , it would be rather imprudent to suggest that the same dominance takes place here without carrying out a parametric study of σ11 similar to that we have undertook for σ22 . To that end, Figs. 9 and 11 show the comparisons between the tensile and compressive peak values of σ11 and σ22 for varying ζ . For the peak compressive stress, σ11 is still very clearly dominated by the σ22 for all ζ , with the former never exceeding 63% of the latter, Fig. 12. For the peak tensile stress, however, the dominance is somewhat less pronounced, with σ11 being as high as almost 75% of σ22 for certain values of ζ , Fig. 10; yet, we are still able to state that the peak longitudinal stress is clearly dominated by the peak transverse stress for all values of ζ . Therefore, we are able to answer ‘Yes’ to the question posed in the title of the this subsection. 5.5. Location and timing of stress peaks Along with the presented results for the values of the peak stress, it is also of interest to analyze where and when the peaks occur (only the transverse stress is addressed in this context, due to its dominance). To that end, we look at the plots summarizing the spatial (angular and axial) and temporal locations of the tensile and compressive peaks, Figs. 13,

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Fig. 8. Ratio of the peak compressive transverse stress produced by the 2D model to that produced by the 3D model.

Fig. 9. Peak tensile transverse stress, red, compared to the peak tensile longitudinal stress, black.

14 and 15, respectively; we note that in order to ensure an adequate appearance of all the potential overlaps, we used both different size and different color for the respective symbols in the plots. The already discussed difference in the phenomenology of the peak tensile stress for the low values of ζ is now particularly apparent, and it is represented by the dramatic changes seen in all three graphs at ζ = 0.286. The axial difference seems to be particularly important from the practical point of view due to the fact that the computational costs of determining the peak stress over the entire shell and those of determining it only in the mid-section x = 0 differ by two orders of magnitude (assuming that the same spatial and temporal resolution is to be maintained), and it

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Fig. 10. Ratio of the peak tensile longitudinal stress to the peak tensile transverse stress.

Fig. 11. Peak compressive transverse stress, red, compared to the peak compressive longitudinal stress, black.

is incomparably more attractive to resort to the mid-section parametric studies whenever possible. It is, therefore, with a relief that one observes that the whole-shell parametric studies are only necessary for the very low values of ζ . The compressive peak is always reached in the mid-section, but its angular location follows a rather irregular pattern, mirrored by the irregularity of the temporal location. These irregularities are due to the overall complexity of the dynamic stress field, but a sudden change in the location of the peak does not necessarily imply a dramatic phenomenological change like the one we have just discussed for the tensile stress — often the phenomenology remains the same, but the peak stress values attributed to a certain phenomenon simply become higher or lower.

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Fig. 12. Ratio of the peak compressive longitudinal stress to the peak compressive transverse stress.

Fig. 13. Angular location of the peaks of σ22 : tensile peak, black, and compressive peak, red.

In order to illustrate this, we address the sudden change of the angular location of the compressive peak (from 0.89 to 2.64) seen when ζ changes from 0.336 (ci = 470 m/s) to 0.343 (ci = 480 m/s), with the corresponding temporal locations being also very different, 0.45 and 1.20, respectively. Fig. 16 shows the relevant regions of the shell for both values of ζ at both instants, and one observes that the stress patterns in question are virtually indistinguishable — it is apparent that the phenomenology of the stress dynamics remains very much the same, and that the differences in the angular and temporal locations of the peaks are attributed simply to the fact that for ζ = 0.336 the highest stress is attained at the earlier instant, while for ζ = 0.343 it is attained at the later one.

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Fig. 14. Axial location of the peaks of σ22 : tensile peak, black, and compressive peak, red.

Fig. 15. Temporal location of the peaks of σ22 : tensile peak, black, and compressive peak, red.

5.6. Summary of computation expenses At the closing of this section, we would like to summarize the computation expenses incurred. All the computations were carried out on the Intel Core i7 3.40 GHz CPU with 16 GB of RAM. For the 2D scenario, computing 151 harmonics of the displacements in the dimensionless time range [0, 10] with the time step of 0.001 took 291 s. Computing the displacement and stress values at a single point took 11 and 24 µs,

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Fig. 16. Stress state of the shell at t = 0.50, top row, and t = 1.20, bottom row, for the case of ζ = 0.336, left column, and ζ = 0.343, right column; the top row shows the front view of the shell, and the bottom row shows the back view of the shell.

respectively, with the extrema search requiring, for a single value of ζ , only a few seconds on any reasonably fine spatial–temporal mesh. The 3D computations were much more costly. Specifically, computing the displacement harmonics for the stated number of terms in the respective series (201 for the axial coordinate and 81 for the angular one) with the time step of 0.00125 in the range [0, 10] took 5.47 h (i.e., almost two orders of magnitude higher computational costs than those incurred by the 2D model). The computation of the displacement and stress values at one point took 1.8 and 6.7 ms, respectively, with the extrema search requiring now many hours for a single value of ζ even on modestly fine meshes. 6. Conclusions We have considered the interaction between an external spherical shock wave and a circular cylindrical shell filled with and submerged into different fluids, and focused our attention on the dynamics of the stress state of the shell and on how the stress state evolves as the sound speed in the internal fluid changes while the sound speed in the external fluid remains constant (the single parameter that drove all the phenomenological changes in the system was the ratio of the internal and external acoustic speeds, ζ ). Of particular interest to us was the evolution of the peak stress, both tensile and compressive. The stress state of the shell was simulated for a typical shock wave with the stand-off of four radii of the shell for several values of ζ , and the sequences of images summarizing the dynamics of the stress state were presented for three most characteristic cases. Then, the parametric studies of the extrema of the transverse and longitudinal stress have been carried out for 0.214 ≤ ζ ≤ 1 (300 ≤ ci ≤ 1400 m/s). The evolution of the peak tensile transverse stress was found to be particularly interesting. Specifically, it was observed that sometimes as small a change of ζ as 3% results in as large a change of the peak tensile stress as 35%. Of perhaps even

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higher importance was the finding that the peak stress values produced by the fully-3D model can be both higher and lower than the respective values produced by the simplified 2D model used in the parent study, Iakovlev et al. (2011); this finding eliminates the possibility of relying on the 2D model for all values of ζ with the expectation that it would provide a good conservative estimate of the peak tensile stress. The evolution of the peak compressive transverse stress was considerably less challenging to analyze, with the changes exhibiting a quasi-linear trend, and with the results produced by the 2D and 3D models being, for all values of ζ , almost exact multiples of each other with the respective factor varying between 1.60 and 1.66; therefore, we are able to state that the 2D model always significantly overestimates the peak compressive transverse stress, and does so in a very predictable manner. The extrema of the transverse stress were compared to those of the longitudinal stress, and the clear dominance of the former was established for all values of ζ - the peak compressive longitudinal stress never exceeded 63% of the peak compressive transverse stress, while the peak tensile longitudinal stress was at most 75% of the peak tensile transverse stress. From the practitioner’s point of view, this finding is of particular importance since it extends what has been observed for ζ ≈ 1 to all ζ values considered here. The spatial and temporal locations of the stress peaks were analyzed as well, and the nature of the sudden changes of the locations was examined. Of particular practical interest was the fact that while the peak compressive transverse stress is always reached in the mid-section of the shell (x = 0), the peak tensile transverse stress can be reached, depending on the value of ζ , both in the mid-section and far away from it; this finding makes it impossible for one to always rely on the outcomes of the parametric studies that are focusing only on the mid-section, with the much more computationally expensive full-shell parametric studies being necessary for certain values of ζ . The values of ζ that yielded the off-midsection peaks were observed to be very low, thus making the scenarios where the internal fluid has very low acoustic speed unique in one more way, in addition to the several hydrodynamic peculiarities that have already been reported for such low-ζ scenarios in the parent study, Iakovlev et al. (2011). 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